Integralgleichung

Exakte DGL

Trennung der Variablen (Separation)

Trennung der Variablen

Anfangswertproblem

Satz von Picard-Lindelöf (Existenz der Lösung)

Satz von Picard-Lindelöf 

Bernoullische DGL

Riccatti-DGL

Riccatische DGL

Darstellungsmatrizen in neuer Basis darstellen

Basis wechsel, Transformationsmatrizen

Monotieverhalten

Cauchy-Produktformel

physik

elastischer stoss

lösbarkeit einer Matrix

kinematik

impuls

kugelkoordinaten une zylinderkoordinaten 

Matrix

kettenregel

Lösbarkeit einer Matrix

Glechgewicht

Schiefe Ebene

Cauchy Verdichtungssatz 

Schaltvorgänge

Singulärwertzerlegung

Spann

Lineares Gleichungssystem

spannung

Basis bestimmen

Fourier-Reihen

Unterschied In/Homogen

Gruppen

lineare abbildung 

quadratische ergänzung

basis 

QR-Zerlegung

Bild & Urbild

Zweigstromanalyse

Transistor als Thermometer

Investment

Teilfolgen

Epsilon Delta Kriterium

prinzip der virtuellen verrückung 

Linear Kombination

Cholesky Zerlegung

Polynomfunktionen beschreiben

Teilchenphysik und Festkörperphysik

test

Regel von de L'Hopital

KRAFT

Grenzwert Funktion 

Folgenkriterium

ϵ-δ Kriterium 

Beweis Stetigkeit 

Basiswechselsatz

Vorderrad fahradgabel lagerkräfte berechnen 

Spannungsregelung

Test

Induktion 

Jordan-Normalform

Inverse einer Matrix

Gleichmäßige Stetigkeit

addition

Stationäre Stoffübertragung

komplexe Ungleichungen 

Rekursive Folgen

Exponentialreihe

restglied

Newton Verfahren

Randwertaufgabe

Numerische Integration: Trapez- und Simpsonregel

dualräume

Dualraum

Komplexe Nullstellen

Chemie

Folgedefinition des Funktionslimes

Grenzwerte und Stetigkeit 

Komplexe Zahlen

Direkte Summe 

Exponentielle Verzinsung

Tilgungsrechnung

Abschreibungsrechnung

Zins- und Barwertrechnung

Grenzkosten

Grenzerlös, Grenzgewinn und Grenzproduktivität

Ökonomische Anwendung

Kostenfunktion

Elementare Funktionen 

Dreisatzrechnung

Vereinfachung von Themen

Summen- und Produktzeichen

Elementargeometrie

Trägheitsmomente

Flächenträgheitsmoment

Komposition

Zentrales, ebenes Kräftesystem

Ableitungsregeln

Baustatik

Gleichförmige geradlinige Bewegung

\begin{align*} y(x)&amp;=\frac{2 \cos (x)}{\sin (x)^{2}+c}+\cos (x) \end{align*}

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Benutze zum Lösen folgenden Zusammenhang: \[ \sin (2 x)=2 \sin (x) \cos (x) \]

Benutze zum Lösen folgenden Zusammenhang: <mjx-container class="MathJax" jax="SVG" display="true" justify="left"><svg alt="
\sin (2 x)=2 \sin (x) \cos (x)
" style="vertical-align: -0.566ex" xmlns="http://www.w3.org/2000/svg" width="23.78ex" height="2.262ex" role="img" focusable="false" viewBox="0 -750 10510.9 1000" xmlns:xlink="http://www.w3.org/1999/xlink"><defs><path id="MJX-12421-TEX-N-73" d="M295 316Q295 356 268 385T190 414Q154 414 128 401Q98 382 98 349Q97 344 98 336T114 312T157 287Q175 282 201 278T245 269T277 256Q294 248 310 236T342 195T359 133Q359 71 321 31T198 -10H190Q138 -10 94 26L86 19L77 10Q71 4 65 -1L54 -11H46H42Q39 -11 33 -5V74V132Q33 153 35 157T45 162H54Q66 162 70 158T75 146T82 119T101 77Q136 26 198 26Q295 26 295 104Q295 133 277 151Q257 175 194 187T111 210Q75 227 54 256T33 318Q33 357 50 384T93 424T143 442T187 447H198Q238 447 268 432L283 424L292 431Q302 440 314 448H322H326Q329 448 335 442V310L329 304H301Q295 310 295 316Z"></path><path id="MJX-12421-TEX-N-69" d="M69 609Q69 637 87 653T131 669Q154 667 171 652T188 609Q188 579 171 564T129 549Q104 549 87 564T69 609ZM247 0Q232 3 143 3Q132 3 106 3T56 1L34 0H26V46H42Q70 46 91 49Q100 53 102 60T104 102V205V293Q104 345 102 359T88 378Q74 385 41 385H30V408Q30 431 32 431L42 432Q52 433 70 434T106 436Q123 437 142 438T171 441T182 442H185V62Q190 52 197 50T232 46H255V0H247Z"></path><path id="MJX-12421-TEX-N-6E" d="M41 46H55Q94 46 102 60V68Q102 77 102 91T102 122T103 161T103 203Q103 234 103 269T102 328V351Q99 370 88 376T43 385H25V408Q25 431 27 431L37 432Q47 433 65 434T102 436Q119 437 138 438T167 441T178 442H181V402Q181 364 182 364T187 369T199 384T218 402T247 421T285 437Q305 442 336 442Q450 438 463 329Q464 322 464 190V104Q464 66 466 59T477 49Q498 46 526 46H542V0H534L510 1Q487 2 460 2T422 3Q319 3 310 0H302V46H318Q379 46 379 62Q380 64 380 200Q379 335 378 343Q372 371 358 385T334 402T308 404Q263 404 229 370Q202 343 195 315T187 232V168V108Q187 78 188 68T191 55T200 49Q221 46 249 46H265V0H257L234 1Q210 2 183 2T145 3Q42 3 33 0H25V46H41Z"></path><path id="MJX-12421-TEX-N-2061" d=""></path><path id="MJX-12421-TEX-N-28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z"></path><path id="MJX-12421-TEX-N-32" d="M109 429Q82 429 66 447T50 491Q50 562 103 614T235 666Q326 666 387 610T449 465Q449 422 429 383T381 315T301 241Q265 210 201 149L142 93L218 92Q375 92 385 97Q392 99 409 186V189H449V186Q448 183 436 95T421 3V0H50V19V31Q50 38 56 46T86 81Q115 113 136 137Q145 147 170 174T204 211T233 244T261 278T284 308T305 340T320 369T333 401T340 431T343 464Q343 527 309 573T212 619Q179 619 154 602T119 569T109 550Q109 549 114 549Q132 549 151 535T170 489Q170 464 154 447T109 429Z"></path><path id="MJX-12421-TEX-I-1D465" d="M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z"></path><path id="MJX-12421-TEX-N-29" d="M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z"></path><path id="MJX-12421-TEX-N-3D" d="M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z"></path><path id="MJX-12421-TEX-N-63" d="M370 305T349 305T313 320T297 358Q297 381 312 396Q317 401 317 402T307 404Q281 408 258 408Q209 408 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Die DGL ist von der Form \[ y^{\prime}+g(x) \cdot y+h(x) \cdot y^{2}=k(x) \]
Wäre $k=0$, wäre es eine Bernoulli-DGL. Allerdings gibt es x-Werte im Definitionsbereich für die $k\neq0$.

Die DGL ist von der Form <mjx-container class="MathJax" jax="SVG" display="true" justify="left"><svg alt="
y^{\prime}+g(x) \cdot y+h(x) \cdot y^{2}=k(x)
" style="vertical-align: -0.566ex" xmlns="http://www.w3.org/2000/svg" width="29.333ex" height="2.565ex" role="img" focusable="false" viewBox="0 -883.9 12965.3 1133.9" xmlns:xlink="http://www.w3.org/1999/xlink"><defs><path id="MJX-12422-TEX-I-1D466" d="M21 287Q21 301 36 335T84 406T158 442Q199 442 224 419T250 355Q248 336 247 334Q247 331 231 288T198 191T182 105Q182 62 196 45T238 27Q261 27 281 38T312 61T339 94Q339 95 344 114T358 173T377 247Q415 397 419 404Q432 431 462 431Q475 431 483 424T494 412T496 403Q496 390 447 193T391 -23Q363 -106 294 -155T156 -205Q111 -205 77 -183T43 -117Q43 -95 50 -80T69 -58T89 -48T106 -45Q150 -45 150 -87Q150 -107 138 -122T115 -142T102 -147L99 -148Q101 -153 118 -160T152 -167H160Q177 -167 186 -165Q219 -156 247 -127T290 -65T313 -9T321 21L315 17Q309 13 296 6T270 -6Q250 -11 231 -11Q185 -11 150 11T104 82Q103 89 103 113Q103 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Z"></path><path id="MJX-12422-TEX-N-2032" d="M79 43Q73 43 52 49T30 61Q30 68 85 293T146 528Q161 560 198 560Q218 560 240 545T262 501Q262 496 260 486Q259 479 173 263T84 45T79 43Z"></path><path id="MJX-12422-TEX-N-2B" d="M56 237T56 250T70 270H369V420L370 570Q380 583 389 583Q402 583 409 568V270H707Q722 262 722 250T707 230H409V-68Q401 -82 391 -82H389H387Q375 -82 369 -68V230H70Q56 237 56 250Z"></path><path id="MJX-12422-TEX-I-1D454" d="M311 43Q296 30 267 15T206 0Q143 0 105 45T66 160Q66 265 143 353T314 442Q361 442 401 394L404 398Q406 401 409 404T418 412T431 419T447 422Q461 422 470 413T480 394Q480 379 423 152T363 -80Q345 -134 286 -169T151 -205Q10 -205 10 -137Q10 -111 28 -91T74 -71Q89 -71 102 -80T116 -111Q116 -121 114 -130T107 -144T99 -154T92 -162L90 -164H91Q101 -167 151 -167Q189 -167 211 -155Q234 -144 254 -122T282 -75Q288 -56 298 -13Q311 35 311 43ZM384 328L380 339Q377 350 375 354T369 368T359 382T346 393T328 402T306 405Q262 405 221 352Q191 313 171 233T151 117Q151 38 213 38Q269 38 323 108L331 118L384 328Z"></path><path id="MJX-12422-TEX-N-28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z"></path><path id="MJX-12422-TEX-I-1D465" d="M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z"></path><path id="MJX-12422-TEX-N-29" d="M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z"></path><path id="MJX-12422-TEX-N-22C5" d="M78 250Q78 274 95 292T138 310Q162 310 180 294T199 251Q199 226 182 208T139 190T96 207T78 250Z"></path><path id="MJX-12422-TEX-I-210E" d="M137 683Q138 683 209 688T282 694Q294 694 294 685Q294 674 258 534Q220 386 220 383Q220 381 227 388Q288 442 357 442Q411 442 444 415T478 336Q478 285 440 178T402 50Q403 36 407 31T422 26Q450 26 474 56T513 138Q516 149 519 151T535 153Q555 153 555 145Q555 144 551 130Q535 71 500 33Q466 -10 419 -10H414Q367 -10 346 17T325 74Q325 90 361 192T398 345Q398 404 354 404H349Q266 404 205 306L198 293L164 158Q132 28 127 16Q114 -11 83 -11Q69 -11 59 -2T48 16Q48 30 121 320L195 616Q195 629 188 632T149 637H128Q122 643 122 645T124 664Q129 683 137 683Z"></path><path id="MJX-12422-TEX-N-32" d="M109 429Q82 429 66 447T50 491Q50 562 103 614T235 666Q326 666 387 610T449 465Q449 422 429 383T381 315T301 241Q265 210 201 149L142 93L218 92Q375 92 385 97Q392 99 409 186V189H449V186Q448 183 436 95T421 3V0H50V19V31Q50 38 56 46T86 81Q115 113 136 137Q145 147 170 174T204 211T233 244T261 278T284 308T305 340T320 369T333 401T340 431T343 464Q343 527 309 573T212 619Q179 619 154 602T119 569T109 550Q109 549 114 549Q132 549 151 535T170 489Q170 464 154 447T109 429Z"></path><path id="MJX-12422-TEX-N-3D" d="M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z"></path><path id="MJX-12422-TEX-I-1D458" d="M121 647Q121 657 125 670T137 683Q138 683 209 688T282 694Q294 694 294 686Q294 679 244 477Q194 279 194 272Q213 282 223 291Q247 309 292 354T362 415Q402 442 438 442Q468 442 485 423T503 369Q503 344 496 327T477 302T456 291T438 288Q418 288 406 299T394 328Q394 353 410 369T442 390L458 393Q446 405 434 405H430Q398 402 367 380T294 316T228 255Q230 254 243 252T267 246T293 238T320 224T342 206T359 180T365 147Q365 130 360 106T354 66Q354 26 381 26Q429 26 459 145Q461 153 479 153H483Q499 153 499 144Q499 139 496 130Q455 -11 378 -11Q333 -11 305 15T277 90Q277 108 280 121T283 145Q283 167 269 183T234 206T200 217T182 220H180Q168 178 159 139T145 81T136 44T129 20T122 7T111 -2Q98 -11 83 -11Q66 -11 57 -1T48 16Q48 26 85 176T158 471L195 616Q196 629 188 632T149 637H144Q134 637 131 637T124 640T121 647Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><g data-mml-node="math"><g data-mml-node="msup"><g data-mml-node="mi"><use xlink:href="#MJX-12422-TEX-I-1D466"></use></g><g data-mml-node="TeXAtom" transform="translate(490, 413) scale(0.707)" data-mjx-texclass="ORD"><g data-mml-node="mi"><use xlink:href="#MJX-12422-TEX-N-2032"></use></g></g></g><g data-mml-node="mo" transform="translate(956.7, 0)"><use xlink:href="#MJX-12422-TEX-N-2B"></use></g><g data-mml-node="mi" transform="translate(1956.9, 0)"><use xlink:href="#MJX-12422-TEX-I-1D454"></use></g><g data-mml-node="mo" transform="translate(2433.9, 0)"><use xlink:href="#MJX-12422-TEX-N-28"></use></g><g data-mml-node="mi" transform="translate(2822.9, 0)"><use xlink:href="#MJX-12422-TEX-I-1D465"></use></g><g data-mml-node="mo" transform="translate(3394.9, 0)"><use xlink:href="#MJX-12422-TEX-N-29"></use></g><g data-mml-node="mo" transform="translate(4006.1, 0)"><use xlink:href="#MJX-12422-TEX-N-22C5"></use></g><g data-mml-node="mi" transform="translate(4506.3, 0)"><use xlink:href="#MJX-12422-TEX-I-1D466"></use></g><g data-mml-node="mo" transform="translate(5218.6, 0)"><use xlink:href="#MJX-12422-TEX-N-2B"></use></g><g data-mml-node="mi" transform="translate(6218.8, 0)"><use xlink:href="#MJX-12422-TEX-I-210E"></use></g><g data-mml-node="mo" transform="translate(6794.8, 0)"><use xlink:href="#MJX-12422-TEX-N-28"></use></g><g data-mml-node="mi" transform="translate(7183.8, 0)"><use xlink:href="#MJX-12422-TEX-I-1D465"></use></g><g data-mml-node="mo" transform="translate(7755.8, 0)"><use xlink:href="#MJX-12422-TEX-N-29"></use></g><g data-mml-node="mo" transform="translate(8367, 0)"><use xlink:href="#MJX-12422-TEX-N-22C5"></use></g><g data-mml-node="msup" transform="translate(8867.2, 0)"><g data-mml-node="mi"><use xlink:href="#MJX-12422-TEX-I-1D466"></use></g><g data-mml-node="TeXAtom" transform="translate(490, 413) scale(0.707)" data-mjx-texclass="ORD"><g data-mml-node="mn"><use xlink:href="#MJX-12422-TEX-N-32"></use></g></g></g><g data-mml-node="mo" transform="translate(10038.6, 0)"><use xlink:href="#MJX-12422-TEX-N-3D"></use></g><g data-mml-node="mi" transform="translate(11094.3, 0)"><use xlink:href="#MJX-12422-TEX-I-1D458"></use></g><g data-mml-node="mo" transform="translate(11615.3, 0)"><use xlink:href="#MJX-12422-TEX-N-28"></use></g><g data-mml-node="mi" transform="translate(12004.3, 0)"><use xlink:href="#MJX-12422-TEX-I-1D465"></use></g><g data-mml-node="mo" transform="translate(12576.3, 0)"><use xlink:href="#MJX-12422-TEX-N-29"></use></g></g></g></svg></mjx-container>
Wäre <mjx-container class="MathJax" jax="SVG"><svg alt="k=0" style="vertical-align: -0.186ex" xmlns="http://www.w3.org/2000/svg" width="5.327ex" height="1.756ex" role="img" focusable="false" viewBox="0 -694 2354.6 776" xmlns:xlink="http://www.w3.org/1999/xlink"><defs><path id="MJX-12423-TEX-I-1D458" d="M121 647Q121 657 125 670T137 683Q138 683 209 688T282 694Q294 694 294 686Q294 679 244 477Q194 279 194 272Q213 282 223 291Q247 309 292 354T362 415Q402 442 438 442Q468 442 485 423T503 369Q503 344 496 327T477 302T456 291T438 288Q418 288 406 299T394 328Q394 353 410 369T442 390L458 393Q446 405 434 405H430Q398 402 367 380T294 316T228 255Q230 254 243 252T267 246T293 238T320 224T342 206T359 180T365 147Q365 130 360 106T354 66Q354 26 381 26Q429 26 459 145Q461 153 479 153H483Q499 153 499 144Q499 139 496 130Q455 -11 378 -11Q333 -11 305 15T277 90Q277 108 280 121T283 145Q283 167 269 183T234 206T200 217T182 220H180Q168 178 159 139T145 81T136 44T129 20T122 7T111 -2Q98 -11 83 -11Q66 -11 57 -1T48 16Q48 26 85 176T158 471L195 616Q196 629 188 632T149 637H144Q134 637 131 637T124 640T121 647Z"></path><path id="MJX-12423-TEX-N-3D" d="M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z"></path><path id="MJX-12423-TEX-N-30" d="M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><g data-mml-node="math"><g data-mml-node="mi"><use xlink:href="#MJX-12423-TEX-I-1D458"></use></g><g data-mml-node="mo" transform="translate(798.8, 0)"><use xlink:href="#MJX-12423-TEX-N-3D"></use></g><g data-mml-node="mn" transform="translate(1854.6, 0)"><use xlink:href="#MJX-12423-TEX-N-30"></use></g></g></g></svg></mjx-container>, wäre es eine Bernoulli-DGL. Allerdings gibt es x-Werte im Definitionsbereich für die <mjx-container class="MathJax" jax="SVG"><svg alt="k\neq0" style="vertical-align: -0.486ex" xmlns="http://www.w3.org/2000/svg" width="5.327ex" height="2.106ex" role="img" focusable="false" viewBox="0 -716 2354.6 931" xmlns:xlink="http://www.w3.org/1999/xlink"><defs><path id="MJX-12424-TEX-I-1D458" d="M121 647Q121 657 125 670T137 683Q138 683 209 688T282 694Q294 694 294 686Q294 679 244 477Q194 279 194 272Q213 282 223 291Q247 309 292 354T362 415Q402 442 438 442Q468 442 485 423T503 369Q503 344 496 327T477 302T456 291T438 288Q418 288 406 299T394 328Q394 353 410 369T442 390L458 393Q446 405 434 405H430Q398 402 367 380T294 316T228 255Q230 254 243 252T267 246T293 238T320 224T342 206T359 180T365 147Q365 130 360 106T354 66Q354 26 381 26Q429 26 459 145Q461 153 479 153H483Q499 153 499 144Q499 139 496 130Q455 -11 378 -11Q333 -11 305 15T277 90Q277 108 280 121T283 145Q283 167 269 183T234 206T200 217T182 220H180Q168 178 159 139T145 81T136 44T129 20T122 7T111 -2Q98 -11 83 -11Q66 -11 57 -1T48 16Q48 26 85 176T158 471L195 616Q196 629 188 632T149 637H144Q134 637 131 637T124 640T121 647Z"></path><path id="MJX-12424-TEX-N-2260" d="M166 -215T159 -215T147 -212T141 -204T139 -197Q139 -190 144 -183L306 133H70Q56 140 56 153Q56 168 72 173H327L406 327H72Q56 332 56 347Q56 360 70 367H426Q597 702 602 707Q605 716 618 716Q625 716 630 712T636 703T638 696Q638 692 471 367H707Q722 359 722 347Q722 336 708 328L451 327L371 173H708Q722 163 722 153Q722 140 707 133H351Q175 -210 170 -212Q166 -215 159 -215Z"></path><path id="MJX-12424-TEX-N-30" d="M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><g data-mml-node="math"><g data-mml-node="mi"><use xlink:href="#MJX-12424-TEX-I-1D458"></use></g><g data-mml-node="mo" transform="translate(798.8, 0)"><use xlink:href="#MJX-12424-TEX-N-2260"></use></g><g data-mml-node="mn" transform="translate(1854.6, 0)"><use xlink:href="#MJX-12424-TEX-N-30"></use></g></g></g></svg></mjx-container>.

Rate eine spezielle Losung $y_{s}$

Rate eine spezielle Losung <mjx-container class="MathJax" jax="SVG"><svg alt="y_{s}" style="vertical-align: -0.464ex" xmlns="http://www.w3.org/2000/svg" width="1.972ex" height="1.464ex" role="img" focusable="false" viewBox="0 -442 871.6 647" xmlns:xlink="http://www.w3.org/1999/xlink"><defs><path id="MJX-12425-TEX-I-1D466" d="M21 287Q21 301 36 335T84 406T158 442Q199 442 224 419T250 355Q248 336 247 334Q247 331 231 288T198 191T182 105Q182 62 196 45T238 27Q261 27 281 38T312 61T339 94Q339 95 344 114T358 173T377 247Q415 397 419 404Q432 431 462 431Q475 431 483 424T494 412T496 403Q496 390 447 193T391 -23Q363 -106 294 -155T156 -205Q111 -205 77 -183T43 -117Q43 -95 50 -80T69 -58T89 -48T106 -45Q150 -45 150 -87Q150 -107 138 -122T115 -142T102 -147L99 -148Q101 -153 118 -160T152 -167H160Q177 -167 186 -165Q219 -156 247 -127T290 -65T313 -9T321 21L315 17Q309 13 296 6T270 -6Q250 -11 231 -11Q185 -11 150 11T104 82Q103 89 103 113Q103 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Z"></path><path id="MJX-12425-TEX-I-1D460" d="M131 289Q131 321 147 354T203 415T300 442Q362 442 390 415T419 355Q419 323 402 308T364 292Q351 292 340 300T328 326Q328 342 337 354T354 372T367 378Q368 378 368 379Q368 382 361 388T336 399T297 405Q249 405 227 379T204 326Q204 301 223 291T278 274T330 259Q396 230 396 163Q396 135 385 107T352 51T289 7T195 -10Q118 -10 86 19T53 87Q53 126 74 143T118 160Q133 160 146 151T160 120Q160 94 142 76T111 58Q109 57 108 57T107 55Q108 52 115 47T146 34T201 27Q237 27 263 38T301 66T318 97T323 122Q323 150 302 164T254 181T195 196T148 231Q131 256 131 289Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><g data-mml-node="math"><g data-mml-node="msub"><g data-mml-node="mi"><use xlink:href="#MJX-12425-TEX-I-1D466"></use></g><g data-mml-node="TeXAtom" transform="translate(490, -150) scale(0.707)" data-mjx-texclass="ORD"><g data-mml-node="mi"><use xlink:href="#MJX-12425-TEX-I-1D460"></use></g></g></g></g></g></svg></mjx-container>

Naheliegend als Lösung wären die Sinusfunktion oder die Kosinusfunktion.

Probiere $\sin(x)$ als spezielle Lösung aus:

Probiere <mjx-container class="MathJax" jax="SVG"><svg alt="\sin(x)" style="vertical-align: -0.566ex" xmlns="http://www.w3.org/2000/svg" width="5.833ex" height="2.262ex" role="img" focusable="false" viewBox="0 -750 2578 1000" xmlns:xlink="http://www.w3.org/1999/xlink"><defs><path id="MJX-12426-TEX-N-73" d="M295 316Q295 356 268 385T190 414Q154 414 128 401Q98 382 98 349Q97 344 98 336T114 312T157 287Q175 282 201 278T245 269T277 256Q294 248 310 236T342 195T359 133Q359 71 321 31T198 -10H190Q138 -10 94 26L86 19L77 10Q71 4 65 -1L54 -11H46H42Q39 -11 33 -5V74V132Q33 153 35 157T45 162H54Q66 162 70 158T75 146T82 119T101 77Q136 26 198 26Q295 26 295 104Q295 133 277 151Q257 175 194 187T111 210Q75 227 54 256T33 318Q33 357 50 384T93 424T143 442T187 447H198Q238 447 268 432L283 424L292 431Q302 440 314 448H322H326Q329 448 335 442V310L329 304H301Q295 310 295 316Z"></path><path id="MJX-12426-TEX-N-69" d="M69 609Q69 637 87 653T131 669Q154 667 171 652T188 609Q188 579 171 564T129 549Q104 549 87 564T69 609ZM247 0Q232 3 143 3Q132 3 106 3T56 1L34 0H26V46H42Q70 46 91 49Q100 53 102 60T104 102V205V293Q104 345 102 359T88 378Q74 385 41 385H30V408Q30 431 32 431L42 432Q52 433 70 434T106 436Q123 437 142 438T171 441T182 442H185V62Q190 52 197 50T232 46H255V0H247Z"></path><path id="MJX-12426-TEX-N-6E" d="M41 46H55Q94 46 102 60V68Q102 77 102 91T102 122T103 161T103 203Q103 234 103 269T102 328V351Q99 370 88 376T43 385H25V408Q25 431 27 431L37 432Q47 433 65 434T102 436Q119 437 138 438T167 441T178 442H181V402Q181 364 182 364T187 369T199 384T218 402T247 421T285 437Q305 442 336 442Q450 438 463 329Q464 322 464 190V104Q464 66 466 59T477 49Q498 46 526 46H542V0H534L510 1Q487 2 460 2T422 3Q319 3 310 0H302V46H318Q379 46 379 62Q380 64 380 200Q379 335 378 343Q372 371 358 385T334 402T308 404Q263 404 229 370Q202 343 195 315T187 232V168V108Q187 78 188 68T191 55T200 49Q221 46 249 46H265V0H257L234 1Q210 2 183 2T145 3Q42 3 33 0H25V46H41Z"></path><path id="MJX-12426-TEX-N-2061" d=""></path><path id="MJX-12426-TEX-N-28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z"></path><path id="MJX-12426-TEX-I-1D465" d="M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z"></path><path id="MJX-12426-TEX-N-29" d="M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><g data-mml-node="math"><g data-mml-node="mi"><use xlink:href="#MJX-12426-TEX-N-73"></use><use xlink:href="#MJX-12426-TEX-N-69" transform="translate(394, 0)"></use><use xlink:href="#MJX-12426-TEX-N-6E" transform="translate(672, 0)"></use></g><g data-mml-node="mo" transform="translate(1228, 0)"><use xlink:href="#MJX-12426-TEX-N-2061"></use></g><g data-mml-node="mo" transform="translate(1228, 0)"><use xlink:href="#MJX-12426-TEX-N-28"></use></g><g data-mml-node="mi" transform="translate(1617, 0)"><use xlink:href="#MJX-12426-TEX-I-1D465"></use></g><g data-mml-node="mo" transform="translate(2189, 0)"><use xlink:href="#MJX-12426-TEX-N-29"></use></g></g></g></svg></mjx-container> als spezielle Lösung aus:

\begin{aligned} y^{\prime}+(\tan (x)-\sin (2 x)) \cdot y+\sin (x) \cdot y^{2} &amp;=-\frac{\sin (2 x) \cos (x)}{2} \\ (\sin (x))^{\prime}+(\tan (x)-\sin (2 x)) \cdot \sin (x)+\sin (x)^{3} &amp;=-\frac{\sin (2 x) \cos (x)}{2} \\ \cos (x)+\tan (x) \sin (x)-\sin (2 x) \sin (x)+\sin (x)^{3} &amp;=-\frac{\sin (2 x) \cos (x)}{2} \end{aligned}

Setze zum Test <mjx-container class="MathJax" jax="SVG"><svg alt="x=0" style="vertical-align: -0.186ex" xmlns="http://www.w3.org/2000/svg" width="5.442ex" height="1.692ex" role="img" focusable="false" viewBox="0 -666 2405.6 748" xmlns:xlink="http://www.w3.org/1999/xlink"><defs><path id="MJX-12428-TEX-I-1D465" d="M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z"></path><path id="MJX-12428-TEX-N-3D" d="M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z"></path><path id="MJX-12428-TEX-N-30" d="M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><g data-mml-node="math"><g data-mml-node="mi"><use xlink:href="#MJX-12428-TEX-I-1D465"></use></g><g data-mml-node="mo" transform="translate(849.8, 0)"><use xlink:href="#MJX-12428-TEX-N-3D"></use></g><g data-mml-node="mn" transform="translate(1905.6, 0)"><use xlink:href="#MJX-12428-TEX-N-30"></use></g></g></g></svg></mjx-container> ein:

\begin{aligned} 1+0 \cdot 0-0 \cdot 0+0^{3} &amp;=-\frac{1 \cdot 0}{2} \\ 1 &amp;=0 \end{aligned}

<mjx-container class="MathJax" jax="SVG"><svg alt="\begin{aligned}
1+0 \cdot 0-0 \cdot 0+0^{3} &=-\frac{1 \cdot 0}{2} \\
1 &=0
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transform="translate(7612.2, 0)"><g data-mml-node="mn"><use xlink:href="#MJX-12429-TEX-N-30"></use></g><g data-mml-node="TeXAtom" transform="translate(500, 413) scale(0.707)" data-mjx-texclass="ORD"><g data-mml-node="mn"><use xlink:href="#MJX-12429-TEX-N-33"></use></g></g></g></g><g data-mml-node="mtd" transform="translate(8515.8, 0)"><g data-mml-node="mi"></g><g data-mml-node="mo" transform="translate(277.8, 0)"><use xlink:href="#MJX-12429-TEX-N-3D"></use></g><g data-mml-node="mo" transform="translate(1333.6, 0)"><use xlink:href="#MJX-12429-TEX-N-2212"></use></g><g data-mml-node="mfrac" transform="translate(2111.6, 0)"><g data-mml-node="mrow" transform="translate(220, 676)"><g data-mml-node="mn"><use xlink:href="#MJX-12429-TEX-N-31"></use></g><g data-mml-node="mo" transform="translate(722.2, 0)"><use xlink:href="#MJX-12429-TEX-N-22C5"></use></g><g data-mml-node="mn" transform="translate(1222.4, 0)"><use xlink:href="#MJX-12429-TEX-N-30"></use></g></g><g data-mml-node="mn" transform="translate(831.2, -686)"><use xlink:href="#MJX-12429-TEX-N-32"></use></g><rect width="1922.4" height="60" x="120" y="220"></rect></g></g></g><g data-mml-node="mtr" transform="translate(0, -1164)"><g data-mml-node="mtd" transform="translate(8015.8, 0)"><g data-mml-node="mn"><use xlink:href="#MJX-12429-TEX-N-31"></use></g></g><g data-mml-node="mtd" transform="translate(8515.8, 0)"><g data-mml-node="mi"></g><g data-mml-node="mo" transform="translate(277.8, 0)"><use xlink:href="#MJX-12429-TEX-N-3D"></use></g><g data-mml-node="mn" transform="translate(1333.6, 0)"><use xlink:href="#MJX-12429-TEX-N-30"></use></g></g></g></g></g></g></svg></mjx-container>

Dies ist ein Widerspruch und somit ist $\sin(x)$ keine Lösung.

Dies ist ein Widerspruch und somit ist <mjx-container class="MathJax" jax="SVG"><svg alt="\sin(x)" style="vertical-align: -0.566ex" xmlns="http://www.w3.org/2000/svg" width="5.833ex" height="2.262ex" role="img" focusable="false" viewBox="0 -750 2578 1000" xmlns:xlink="http://www.w3.org/1999/xlink"><defs><path id="MJX-12430-TEX-N-73" d="M295 316Q295 356 268 385T190 414Q154 414 128 401Q98 382 98 349Q97 344 98 336T114 312T157 287Q175 282 201 278T245 269T277 256Q294 248 310 236T342 195T359 133Q359 71 321 31T198 -10H190Q138 -10 94 26L86 19L77 10Q71 4 65 -1L54 -11H46H42Q39 -11 33 -5V74V132Q33 153 35 157T45 162H54Q66 162 70 158T75 146T82 119T101 77Q136 26 198 26Q295 26 295 104Q295 133 277 151Q257 175 194 187T111 210Q75 227 54 256T33 318Q33 357 50 384T93 424T143 442T187 447H198Q238 447 268 432L283 424L292 431Q302 440 314 448H322H326Q329 448 335 442V310L329 304H301Q295 310 295 316Z"></path><path id="MJX-12430-TEX-N-69" d="M69 609Q69 637 87 653T131 669Q154 667 171 652T188 609Q188 579 171 564T129 549Q104 549 87 564T69 609ZM247 0Q232 3 143 3Q132 3 106 3T56 1L34 0H26V46H42Q70 46 91 49Q100 53 102 60T104 102V205V293Q104 345 102 359T88 378Q74 385 41 385H30V408Q30 431 32 431L42 432Q52 433 70 434T106 436Q123 437 142 438T171 441T182 442H185V62Q190 52 197 50T232 46H255V0H247Z"></path><path id="MJX-12430-TEX-N-6E" d="M41 46H55Q94 46 102 60V68Q102 77 102 91T102 122T103 161T103 203Q103 234 103 269T102 328V351Q99 370 88 376T43 385H25V408Q25 431 27 431L37 432Q47 433 65 434T102 436Q119 437 138 438T167 441T178 442H181V402Q181 364 182 364T187 369T199 384T218 402T247 421T285 437Q305 442 336 442Q450 438 463 329Q464 322 464 190V104Q464 66 466 59T477 49Q498 46 526 46H542V0H534L510 1Q487 2 460 2T422 3Q319 3 310 0H302V46H318Q379 46 379 62Q380 64 380 200Q379 335 378 343Q372 371 358 385T334 402T308 404Q263 404 229 370Q202 343 195 315T187 232V168V108Q187 78 188 68T191 55T200 49Q221 46 249 46H265V0H257L234 1Q210 2 183 2T145 3Q42 3 33 0H25V46H41Z"></path><path id="MJX-12430-TEX-N-2061" d=""></path><path id="MJX-12430-TEX-N-28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z"></path><path id="MJX-12430-TEX-I-1D465" d="M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z"></path><path id="MJX-12430-TEX-N-29" d="M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><g data-mml-node="math"><g data-mml-node="mi"><use xlink:href="#MJX-12430-TEX-N-73"></use><use xlink:href="#MJX-12430-TEX-N-69" transform="translate(394, 0)"></use><use xlink:href="#MJX-12430-TEX-N-6E" transform="translate(672, 0)"></use></g><g data-mml-node="mo" transform="translate(1228, 0)"><use xlink:href="#MJX-12430-TEX-N-2061"></use></g><g data-mml-node="mo" transform="translate(1228, 0)"><use xlink:href="#MJX-12430-TEX-N-28"></use></g><g data-mml-node="mi" transform="translate(1617, 0)"><use xlink:href="#MJX-12430-TEX-I-1D465"></use></g><g data-mml-node="mo" transform="translate(2189, 0)"><use xlink:href="#MJX-12430-TEX-N-29"></use></g></g></g></svg></mjx-container> keine Lösung.

Probiere $\cos(x)$ als spezielle Lösung aus:

Probiere <mjx-container class="MathJax" jax="SVG"><svg alt="\cos(x)" style="vertical-align: -0.566ex" xmlns="http://www.w3.org/2000/svg" width="6.081ex" height="2.262ex" role="img" focusable="false" viewBox="0 -750 2688 1000" xmlns:xlink="http://www.w3.org/1999/xlink"><defs><path id="MJX-12431-TEX-N-63" d="M370 305T349 305T313 320T297 358Q297 381 312 396Q317 401 317 402T307 404Q281 408 258 408Q209 408 178 376Q131 329 131 219Q131 137 162 90Q203 29 272 29Q313 29 338 55T374 117Q376 125 379 127T395 129H409Q415 123 415 120Q415 116 411 104T395 71T366 33T318 2T249 -11Q163 -11 99 53T34 214Q34 318 99 383T250 448T370 421T404 357Q404 334 387 320Z"></path><path id="MJX-12431-TEX-N-6F" d="M28 214Q28 309 93 378T250 448Q340 448 405 380T471 215Q471 120 407 55T250 -10Q153 -10 91 57T28 214ZM250 30Q372 30 372 193V225V250Q372 272 371 288T364 326T348 362T317 390T268 410Q263 411 252 411Q222 411 195 399Q152 377 139 338T126 246V226Q126 130 145 91Q177 30 250 30Z"></path><path id="MJX-12431-TEX-N-73" d="M295 316Q295 356 268 385T190 414Q154 414 128 401Q98 382 98 349Q97 344 98 336T114 312T157 287Q175 282 201 278T245 269T277 256Q294 248 310 236T342 195T359 133Q359 71 321 31T198 -10H190Q138 -10 94 26L86 19L77 10Q71 4 65 -1L54 -11H46H42Q39 -11 33 -5V74V132Q33 153 35 157T45 162H54Q66 162 70 158T75 146T82 119T101 77Q136 26 198 26Q295 26 295 104Q295 133 277 151Q257 175 194 187T111 210Q75 227 54 256T33 318Q33 357 50 384T93 424T143 442T187 447H198Q238 447 268 432L283 424L292 431Q302 440 314 448H322H326Q329 448 335 442V310L329 304H301Q295 310 295 316Z"></path><path id="MJX-12431-TEX-N-2061" d=""></path><path id="MJX-12431-TEX-N-28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z"></path><path id="MJX-12431-TEX-I-1D465" d="M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z"></path><path id="MJX-12431-TEX-N-29" d="M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><g data-mml-node="math"><g data-mml-node="mi"><use xlink:href="#MJX-12431-TEX-N-63"></use><use xlink:href="#MJX-12431-TEX-N-6F" transform="translate(444, 0)"></use><use xlink:href="#MJX-12431-TEX-N-73" transform="translate(944, 0)"></use></g><g data-mml-node="mo" transform="translate(1338, 0)"><use xlink:href="#MJX-12431-TEX-N-2061"></use></g><g data-mml-node="mo" transform="translate(1338, 0)"><use xlink:href="#MJX-12431-TEX-N-28"></use></g><g data-mml-node="mi" transform="translate(1727, 0)"><use xlink:href="#MJX-12431-TEX-I-1D465"></use></g><g data-mml-node="mo" transform="translate(2299, 0)"><use xlink:href="#MJX-12431-TEX-N-29"></use></g></g></g></svg></mjx-container> als spezielle Lösung aus:

\begin{aligned} (\cos (x))^{\prime}+(\tan (x)-\sin (2 x)) \cdot \cos (x)+\sin (x) \cos (x)^{2} &amp;=-\frac{\sin (2 x) \cos (x)}{2} \\ -\sin (x)+\tan (x) \cos (x)-\sin (2 x) \cos (x)+\sin (x) \cos (x)^{2} &amp;=-\frac{\sin (2 x) \cos (x)}{2} \\ -\sin (x)+\frac{\sin (x)}{\cos (x)} \cdot \cos (x)-\sin (2 x) \cos (x)+\sin (x) \cos (x)^{2} &amp;=-\frac{\sin (2 x) \cos (x)}{2} \\ -\sin (2 x) \cos (x)+\frac{\sin (2 x) \cos (x)}{2} &amp;=-\frac{\sin (2 x) \cos (x)}{2} \end{aligned}

Diese Gleichung gilt für alle $x$. Somit ist die spezielle Lösung: $y_{s}(x)=\cos (x)$

Diese Gleichung gilt für alle <mjx-container class="MathJax" jax="SVG"><svg alt="x" style="vertical-align: -0.025ex" xmlns="http://www.w3.org/2000/svg" width="1.294ex" height="1.025ex" role="img" focusable="false" viewBox="0 -442 572 453" xmlns:xlink="http://www.w3.org/1999/xlink"><defs><path id="MJX-12433-TEX-I-1D465" d="M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><g data-mml-node="math"><g data-mml-node="mi"><use xlink:href="#MJX-12433-TEX-I-1D465"></use></g></g></g></svg></mjx-container>. Somit ist die spezielle Lösung: <mjx-container class="MathJax" jax="SVG"><svg alt="y_{s}(x)=\cos (x)" style="vertical-align: -0.566ex" xmlns="http://www.w3.org/2000/svg" width="14.125ex" height="2.262ex" role="img" focusable="false" viewBox="0 -750 6243.2 1000" xmlns:xlink="http://www.w3.org/1999/xlink"><defs><path id="MJX-12434-TEX-I-1D466" d="M21 287Q21 301 36 335T84 406T158 442Q199 442 224 419T250 355Q248 336 247 334Q247 331 231 288T198 191T182 105Q182 62 196 45T238 27Q261 27 281 38T312 61T339 94Q339 95 344 114T358 173T377 247Q415 397 419 404Q432 431 462 431Q475 431 483 424T494 412T496 403Q496 390 447 193T391 -23Q363 -106 294 -155T156 -205Q111 -205 77 -183T43 -117Q43 -95 50 -80T69 -58T89 -48T106 -45Q150 -45 150 -87Q150 -107 138 -122T115 -142T102 -147L99 -148Q101 -153 118 -160T152 -167H160Q177 -167 186 -165Q219 -156 247 -127T290 -65T313 -9T321 21L315 17Q309 13 296 6T270 -6Q250 -11 231 -11Q185 -11 150 11T104 82Q103 89 103 113Q103 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Z"></path><path id="MJX-12434-TEX-I-1D460" d="M131 289Q131 321 147 354T203 415T300 442Q362 442 390 415T419 355Q419 323 402 308T364 292Q351 292 340 300T328 326Q328 342 337 354T354 372T367 378Q368 378 368 379Q368 382 361 388T336 399T297 405Q249 405 227 379T204 326Q204 301 223 291T278 274T330 259Q396 230 396 163Q396 135 385 107T352 51T289 7T195 -10Q118 -10 86 19T53 87Q53 126 74 143T118 160Q133 160 146 151T160 120Q160 94 142 76T111 58Q109 57 108 57T107 55Q108 52 115 47T146 34T201 27Q237 27 263 38T301 66T318 97T323 122Q323 150 302 164T254 181T195 196T148 231Q131 256 131 289Z"></path><path id="MJX-12434-TEX-N-28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z"></path><path id="MJX-12434-TEX-I-1D465" d="M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z"></path><path id="MJX-12434-TEX-N-29" d="M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z"></path><path id="MJX-12434-TEX-N-3D" d="M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z"></path><path id="MJX-12434-TEX-N-63" d="M370 305T349 305T313 320T297 358Q297 381 312 396Q317 401 317 402T307 404Q281 408 258 408Q209 408 178 376Q131 329 131 219Q131 137 162 90Q203 29 272 29Q313 29 338 55T374 117Q376 125 379 127T395 129H409Q415 123 415 120Q415 116 411 104T395 71T366 33T318 2T249 -11Q163 -11 99 53T34 214Q34 318 99 383T250 448T370 421T404 357Q404 334 387 320Z"></path><path id="MJX-12434-TEX-N-6F" d="M28 214Q28 309 93 378T250 448Q340 448 405 380T471 215Q471 120 407 55T250 -10Q153 -10 91 57T28 214ZM250 30Q372 30 372 193V225V250Q372 272 371 288T364 326T348 362T317 390T268 410Q263 411 252 411Q222 411 195 399Q152 377 139 338T126 246V226Q126 130 145 91Q177 30 250 30Z"></path><path id="MJX-12434-TEX-N-73" d="M295 316Q295 356 268 385T190 414Q154 414 128 401Q98 382 98 349Q97 344 98 336T114 312T157 287Q175 282 201 278T245 269T277 256Q294 248 310 236T342 195T359 133Q359 71 321 31T198 -10H190Q138 -10 94 26L86 19L77 10Q71 4 65 -1L54 -11H46H42Q39 -11 33 -5V74V132Q33 153 35 157T45 162H54Q66 162 70 158T75 146T82 119T101 77Q136 26 198 26Q295 26 295 104Q295 133 277 151Q257 175 194 187T111 210Q75 227 54 256T33 318Q33 357 50 384T93 424T143 442T187 447H198Q238 447 268 432L283 424L292 431Q302 440 314 448H322H326Q329 448 335 442V310L329 304H301Q295 310 295 316Z"></path><path id="MJX-12434-TEX-N-2061" d=""></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><g data-mml-node="math"><g data-mml-node="msub"><g data-mml-node="mi"><use xlink:href="#MJX-12434-TEX-I-1D466"></use></g><g data-mml-node="TeXAtom" transform="translate(490, -150) scale(0.707)" data-mjx-texclass="ORD"><g data-mml-node="mi"><use xlink:href="#MJX-12434-TEX-I-1D460"></use></g></g></g><g data-mml-node="mo" transform="translate(871.6, 0)"><use xlink:href="#MJX-12434-TEX-N-28"></use></g><g data-mml-node="mi" transform="translate(1260.6, 0)"><use xlink:href="#MJX-12434-TEX-I-1D465"></use></g><g data-mml-node="mo" transform="translate(1832.6, 0)"><use xlink:href="#MJX-12434-TEX-N-29"></use></g><g data-mml-node="mo" transform="translate(2499.4, 0)"><use xlink:href="#MJX-12434-TEX-N-3D"></use></g><g data-mml-node="mi" transform="translate(3555.2, 0)"><use xlink:href="#MJX-12434-TEX-N-63"></use><use xlink:href="#MJX-12434-TEX-N-6F" transform="translate(444, 0)"></use><use xlink:href="#MJX-12434-TEX-N-73" transform="translate(944, 0)"></use></g><g data-mml-node="mo" transform="translate(4893.2, 0)"><use xlink:href="#MJX-12434-TEX-N-2061"></use></g><g data-mml-node="mo" transform="translate(4893.2, 0)"><use xlink:href="#MJX-12434-TEX-N-28"></use></g><g data-mml-node="mi" transform="translate(5282.2, 0)"><use xlink:href="#MJX-12434-TEX-I-1D465"></use></g><g data-mml-node="mo" transform="translate(5854.2, 0)"><use xlink:href="#MJX-12434-TEX-N-29"></use></g></g></g></svg></mjx-container>

Setze $y(x)=v(x)+y_{s}(x)=v(x)+\cos(x)$ in die linke Seite der DGL ein:

Setze <mjx-container class="MathJax" jax="SVG"><svg alt="y(x)=v(x)+y_{s}(x)=v(x)+\cos(x)" style="vertical-align: -0.566ex" xmlns="http://www.w3.org/2000/svg" width="35.139ex" height="2.262ex" role="img" focusable="false" viewBox="0 -750 15531.6 1000" xmlns:xlink="http://www.w3.org/1999/xlink"><defs><path id="MJX-12435-TEX-I-1D466" d="M21 287Q21 301 36 335T84 406T158 442Q199 442 224 419T250 355Q248 336 247 334Q247 331 231 288T198 191T182 105Q182 62 196 45T238 27Q261 27 281 38T312 61T339 94Q339 95 344 114T358 173T377 247Q415 397 419 404Q432 431 462 431Q475 431 483 424T494 412T496 403Q496 390 447 193T391 -23Q363 -106 294 -155T156 -205Q111 -205 77 -183T43 -117Q43 -95 50 -80T69 -58T89 -48T106 -45Q150 -45 150 -87Q150 -107 138 -122T115 -142T102 -147L99 -148Q101 -153 118 -160T152 -167H160Q177 -167 186 -165Q219 -156 247 -127T290 -65T313 -9T321 21L315 17Q309 13 296 6T270 -6Q250 -11 231 -11Q185 -11 150 11T104 82Q103 89 103 113Q103 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Z"></path><path id="MJX-12435-TEX-N-28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z"></path><path id="MJX-12435-TEX-I-1D465" d="M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 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transform="translate(14181.6, 0)"><use xlink:href="#MJX-12435-TEX-N-28"></use></g><g data-mml-node="mi" transform="translate(14570.6, 0)"><use xlink:href="#MJX-12435-TEX-I-1D465"></use></g><g data-mml-node="mo" transform="translate(15142.6, 0)"><use xlink:href="#MJX-12435-TEX-N-29"></use></g></g></g></svg></mjx-container> in die linke Seite der DGL ein:

\begin{aligned} y^{\prime}&amp;+(\tan (x)-\sin (2 x)) \cdot y+\sin (x) \cdot y^{2}\\ &amp;=v^{\prime}-\sin (x)+(\tan (x)-\sin (2 x)) \cdot v+\tan (x) \cos (x)-\sin (2 x) \cos (x)+\sin (x) \cdot(v+\cos (x))^{2} \\ &amp;=v^{\prime}+(\tan (x)-\sin (2 x)) \cdot v-\sin (2 x) \cos (x)+\sin (x)\left(v^{2}+2 \cos (x) \cdot v+\cos (x)^{2}\right) \\
&amp;=v^{\prime}+(\tan (x)-\sin (2 x)) \cdot v-\frac{2 \sin (2 x) \cos (x)}{2}+\sin (x) \cdot v^{2}+\sin (2 x) \cdot v+\frac{\sin (2 x) \cos (x)}{2} \end{aligned}

Füge die rechte Seite der DGL hinzu:

\begin{aligned} v^{\prime}+(\tan (x)-\sin (2 x)) \cdot v+\sin (x) \cdot v^{2}+\sin (2 x) \cdot v &amp;=0 \\ v^{\prime}+\tan (x) \cdot v+\sin (x) \cdot v^{2} &amp;=0 \end{aligned}

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v^{\prime}+(\tan (x)-\sin (2 x)) \cdot v+\sin (x) \cdot v^{2}+\sin (2 x) \cdot v &=0 \\
v^{\prime}+\tan (x) \cdot v+\sin (x) \cdot v^{2} &=0
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Dies ist nun eine Bernoullische DGL.

Transformiere den Ausdruck mit $v=z^{\frac{1}{1-a}}=z^{-1}$

Transformiere den Ausdruck mit <mjx-container class="MathJax" jax="SVG"><svg alt="v=z^{\frac{1}{1-a}}=z^{-1}" style="vertical-align: -0.186ex" xmlns="http://www.w3.org/2000/svg" width="14.254ex" height="2.464ex" role="img" focusable="false" viewBox="0 -1007.1 6300.4 1089.1" xmlns:xlink="http://www.w3.org/1999/xlink"><defs><path id="MJX-12438-TEX-I-1D463" d="M173 380Q173 405 154 405Q130 405 104 376T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Q21 294 29 316T53 368T97 419T160 441Q202 441 225 417T249 361Q249 344 246 335Q246 329 231 291T200 202T182 113Q182 86 187 69Q200 26 250 26Q287 26 319 60T369 139T398 222T409 277Q409 300 401 317T383 343T365 361T357 383Q357 405 376 424T417 443Q436 443 451 425T467 367Q467 340 455 284T418 159T347 40T241 -11Q177 -11 139 22Q102 54 102 117Q102 148 110 181T151 298Q173 362 173 380Z"></path><path id="MJX-12438-TEX-N-3D" d="M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z"></path><path id="MJX-12438-TEX-I-1D467" d="M347 338Q337 338 294 349T231 360Q211 360 197 356T174 346T162 335T155 324L153 320Q150 317 138 317Q117 317 117 325Q117 330 120 339Q133 378 163 406T229 440Q241 442 246 442Q271 442 291 425T329 392T367 375Q389 375 411 408T434 441Q435 442 449 442H462Q468 436 468 434Q468 430 463 420T449 399T432 377T418 358L411 349Q368 298 275 214T160 106L148 94L163 93Q185 93 227 82T290 71Q328 71 360 90T402 140Q406 149 409 151T424 153Q443 153 443 143Q443 138 442 134Q425 72 376 31T278 -11Q252 -11 232 6T193 40T155 57Q111 57 76 -3Q70 -11 59 -11H54H41Q35 -5 35 -2Q35 13 93 84Q132 129 225 214T340 322Q352 338 347 338Z"></path><path id="MJX-12438-TEX-N-31" d="M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z"></path><path id="MJX-12438-TEX-N-2212" d="M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z"></path><path id="MJX-12438-TEX-I-1D44E" d="M33 157Q33 258 109 349T280 441Q331 441 370 392Q386 422 416 422Q429 422 439 414T449 394Q449 381 412 234T374 68Q374 43 381 35T402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487Q506 153 506 144Q506 138 501 117T481 63T449 13Q436 0 417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157ZM351 328Q351 334 346 350T323 385T277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q217 26 254 59T298 110Q300 114 325 217T351 328Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><g data-mml-node="math"><g data-mml-node="mi"><use xlink:href="#MJX-12438-TEX-I-1D463"></use></g><g data-mml-node="mo" transform="translate(762.8, 0)"><use xlink:href="#MJX-12438-TEX-N-3D"></use></g><g data-mml-node="msup" transform="translate(1818.6, 0)"><g data-mml-node="mi"><use xlink:href="#MJX-12438-TEX-I-1D467"></use></g><g data-mml-node="TeXAtom" transform="translate(465, 395.5) scale(0.707)" data-mjx-texclass="ORD"><g data-mml-node="mfrac"><g data-mml-node="mn" transform="translate(682.1, 394) scale(0.707)"><use xlink:href="#MJX-12438-TEX-N-31"></use></g><g data-mml-node="mrow" transform="translate(220, -345) scale(0.707)"><g data-mml-node="mn"><use xlink:href="#MJX-12438-TEX-N-31"></use></g><g data-mml-node="mo" transform="translate(500, 0)"><use xlink:href="#MJX-12438-TEX-N-2212"></use></g><g data-mml-node="mi" transform="translate(1278, 0)"><use xlink:href="#MJX-12438-TEX-I-1D44E"></use></g></g><rect width="1477.7" height="60" x="120" y="220"></rect></g></g></g><g data-mml-node="mo" transform="translate(3826, 0)"><use xlink:href="#MJX-12438-TEX-N-3D"></use></g><g data-mml-node="msup" transform="translate(4881.7, 0)"><g data-mml-node="mi"><use xlink:href="#MJX-12438-TEX-I-1D467"></use></g><g data-mml-node="TeXAtom" transform="translate(465, 363) scale(0.707)" data-mjx-texclass="ORD"><g data-mml-node="mo"><use xlink:href="#MJX-12438-TEX-N-2212"></use></g><g data-mml-node="mn" transform="translate(778, 0)"><use xlink:href="#MJX-12438-TEX-N-31"></use></g></g></g></g></g></svg></mjx-container>

\begin{aligned} v^{\prime}+\tan (x) \cdot v+\sin (x) \cdot v^{2} &amp;=0 \\ \left(z^{-1}\right)^{\prime}+\tan (x) \cdot z^{-1}+\sin (x) \cdot\left(z^{-1}\right)^{2} &amp;=0 \\ -z^{-2} \cdot z^{\prime}+\tan (x) \cdot z^{-1}+\sin (x) \cdot z^{-2} &amp;=0 \\ z^{\prime}-\tan (x) \cdot z-\sin (x) &amp;=0 \end{aligned}

Für DGLs dieser Form gibt es eine Lösungsformel mit $A(x)=\int a(x) \mathrm{d} x$ :

Für DGLs dieser Form gibt es eine Lösungsformel mit <mjx-container class="MathJax" jax="SVG"><svg alt="A(x)=\int a(x) \mathrm{d} x" style="vertical-align: -0.691ex" xmlns="http://www.w3.org/2000/svg" width="16.329ex" height="2.514ex" role="img" focusable="false" viewBox="0 -805.5 7217.2 1111" xmlns:xlink="http://www.w3.org/1999/xlink"><defs><path id="MJX-12440-TEX-I-1D434" d="M208 74Q208 50 254 46Q272 46 272 35Q272 34 270 22Q267 8 264 4T251 0Q249 0 239 0T205 1T141 2Q70 2 50 0H42Q35 7 35 11Q37 38 48 46H62Q132 49 164 96Q170 102 345 401T523 704Q530 716 547 716H555H572Q578 707 578 706L606 383Q634 60 636 57Q641 46 701 46Q726 46 726 36Q726 34 723 22Q720 7 718 4T704 0Q701 0 690 0T651 1T578 2Q484 2 455 0H443Q437 6 437 9T439 27Q443 40 445 43L449 46H469Q523 49 533 63L521 213H283L249 155Q208 86 208 74ZM516 260Q516 271 504 416T490 562L463 519Q447 492 400 412L310 260L413 259Q516 259 516 260Z"></path><path id="MJX-12440-TEX-N-28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 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xlink:href="#MJX-12440-TEX-N-28"></use></g><g data-mml-node="mi" transform="translate(5128.2, 0)"><use xlink:href="#MJX-12440-TEX-I-1D465"></use></g><g data-mml-node="mo" transform="translate(5700.2, 0)"><use xlink:href="#MJX-12440-TEX-N-29"></use></g><g data-mml-node="TeXAtom" data-mjx-texclass="ORD" transform="translate(6089.2, 0)"><g data-mml-node="mi"><use xlink:href="#MJX-12440-TEX-N-64"></use></g></g><g data-mml-node="mi" transform="translate(6645.2, 0)"><use xlink:href="#MJX-12440-TEX-I-1D465"></use></g></g></g></svg></mjx-container> :

\[ \begin{array}{r} z^{\prime}+a(x) \cdot z=b(x) \\ \Rightarrow z(x)=\left(\int b(x) \cdot e^{A(x)} \mathrm{d} x+c\right) \cdot e^{-A(z)} \end{array} \]

<mjx-container class="MathJax" jax="SVG" display="true" justify="left"><svg alt="
\begin{array}{r}
z^{\prime}+a(x) \cdot z=b(x) \\
\Rightarrow z(x)=\left(\int b(x) \cdot e^{A(x)} \mathrm{d} x+c\right) \cdot e^{-A(z)}
\end{array}
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\begin{aligned} A(x) &amp;=\int-\tan (x) \mathrm{d} x \\ &amp;=\int \frac{-\sin (x)}{\cos (x)} \mathrm{d} x \\ &amp;=\ln |\cos (x)| \\ &amp;=\ln (\cos (x)) \end{aligned}

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A(x) &=\int-\tan (x) \mathrm{d} x \\
&=\int \frac{-\sin (x)}{\cos (x)} \mathrm{d} x \\
&=\ln |\cos (x)| \\
&=\ln (\cos (x))
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\begin{aligned} z(x) &amp;=\left(\int b(x) \cdot e^{\ln (\cos (x))} \mathrm{d} x+c\right) \cdot e^{-\ln (\cos (x))} \\ &amp;=\left(\int \sin (x) \cdot \cos (x) \mathrm{d} x+c\right) \cdot \frac{1}{\cos (x)} \\ &amp;=\left(\frac{\sin (x)^{2}}{2}+c\right) \frac{1}{\cos (x)} \\ &amp;=\frac{\sin (x)^{2}+c}{2 \cos (x)} \end{aligned}

\begin{aligned} v(x) &amp;=\frac{1}{z(x)} \\ &amp;=\frac{2 \cos (x)}{\sin (x)^{2}+c} \end{aligned}

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\begin{align*} y(x)&amp;=v(x)+\cos (x) \\ &amp;=\frac{2 \cos (x)}{\sin (x)^{2}+c}+\cos (x) \end{align*}

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Bestimme die allgemeine Lösung der Differentialgleichung: \[ y^{\prime}+(\tan (x)-\sin (2 x)) \cdot y+\sin (x) \cdot y^{2}=-\frac{\sin (2 x) \cos (x)}{2} \]
 
Dabei gelte $x \in\left(-\frac{\pi}{2}, \frac{\pi}{2}\right) .$

Bestimme die allgemeine Lösung der Differentialgleichung: <mjx-container class="MathJax" jax="SVG" display="true" justify="left"><svg alt="
y^{\prime}+(\tan (x)-\sin (2 x)) \cdot y+\sin (x) \cdot y^{2}=-\frac{\sin (2 x) \cos (x)}{2}
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Dies ist eine interaktive Aufgabe zu: Riccatti-DGL mit praktischen Tipps zum Lösen und einer Zusammenfassung der nötigen Theorie

Nicht lineare DGL - Riccatti-DGL | Aufgabe mit Lösung

<h2 class="content_heading">Gewöhnliche DGL Lösungsansätze Übersicht</h2>
Separierbare DGL 1. Ordnung 
Form: $y^{\prime}=f(x) \cdot g(y)$
Lösung mithilfe Trennung der Variablen:
$$\int \frac{1}{g(y)} d y=\int f(x) d x$$
 
Durch Substitution lösbare DGL Form: $\mathrm{y}^{\prime}=\mathrm{f}(\mathrm{ax}+\mathrm{by}+\mathrm{c})$ mit $b \neq 0$ Lösung durch Substitution und Trennung der Variablen:
Substituiere: $u=a x+b y+c$, somit ist $y^{\prime}=f(u)$ Dann ist $u^{\prime}=\frac{d u}{d x}=a+b \frac{d y}{d x}=1 \cdot(a+b \cdot f(u))$ Durch Trennung der Variablen erhältst du die Lösung von $u(x)$. Die Rücksubstitution liefert dir dann $y(x)$
 
<h2 class="content_heading">Lineare DGLs</h2>
Die allgemeine Lösung einer inhomogenen linearen DGL setzt sich aus  1. der allgemeinen Lösung $\mathrm{y}_{\mathrm{h}}(\mathrm{x})$ der zugehörigen homogenen DGL 2. der partikulären Lösung $\mathrm{y}_{\mathrm{p}}(\mathrm{x})$ der inhomogenen DGL zusammen:  $\mathrm{y}(\mathrm{x})=\mathrm{y}_{\mathrm{h}}(\mathrm{x})+\mathrm{y}_{\mathrm{p}}(\mathrm{x})$
 
Homogene lineare DGL 1. Ordnung Form: $\mathrm{y}^{\prime}+\mathrm{f}(\mathrm{x}) \cdot \mathrm{y}=0$ Die allgemeine Lösung lautet:
$\mathrm{y}(\mathrm{x})=\mathrm{C} \cdot \mathrm{e}^{-\mathrm{F}(\mathrm{x})}$, wobei $\mathrm{F}(\mathrm{x})=\int \mathrm{f}(\mathrm{x}) \mathrm{d} \mathrm{x}$ und $\mathrm{C} \in \mathrm{R}$.
 
Inhomogene lineare DGL 1. Ordnung Form: $\mathrm{y}^{\prime}+\mathrm{a}(\mathrm{x}) \cdot \mathrm{y}=\mathrm{b}(\mathrm{x})$ Lösung durch Variation der Konstanten:
$\mathrm{y}(\mathrm{x})=\mathrm{c}(\mathrm{x}) \cdot \mathrm{e}^{-\mathrm{A}(\mathrm{x})}$, wobei $c(x)=\int b(x) \cdot e^{+A(x)} d x$ und $\mathrm{A}(\mathrm{x})=\int \mathrm{a}(\mathrm{x}) \mathrm{d} \mathrm{x}$
 
Inhomogene lineare DGL 1. Ordnung mit konstanten Koeffizienten Form: $y^{\prime}+a_{0} y=g(x)$, wobei $a_{0}=konstant$ Allgemeine Lösung der homogenen DGL:
$\mathrm{y}_{\mathrm{h}}(\mathrm{x})=\mathrm{C} \cdot \mathrm{e}^{-\mathrm{a}_{0} \mathrm{x}}$
 
Partikuläre Lösung der inhomogenen DGL: 
<ol>
<li>Wenn $g(x)$ von der Form: $b_{0}+b_{1} x+b_{2} x^{2}+\ldots+b_{n} x^{n}$ Ansatz: $\mathbf{y}_{\mathbf{p}}(\mathbf{x})= c_{o}+c_{1} x+c_{2} x^{2}+\ldots+c_{n} x^{n}$ </li>
<li>Wenn $g(x)$ von der Form: $b \cdot e^{\alpha x}$ und $\alpha \neq-\mathrm{a}_{\mathrm{o}}$ Ansatz: $\mathbf{y}_{\mathbf{p}}(\mathbf{x})=c \cdot e^{\alpha x}$ </li>
<li>Wenn $g(x)$ von der Form: $b \cdot e^{\alpha x}$ und $\alpha=-a_{0}$ Ansatz: $\mathbf{y}_{\mathbf{p}}(\mathbf{x})=c \cdot x \cdot e^{\alpha x}$ </li>
<li>Wenn $g(x)$ von der Form: $b_{1} \cdot \sin (\beta x)+b_{2} \cdot \cos (\beta x)$ Ansatz: $\mathbf{y}_{\mathbf{p}}(\mathbf{x})= c_{1} \cdot \sin (\beta x)+c_{2} \cdot \cos (\beta x)$</li>
</ol>
Die allgemeine Lösung ist dann: $\mathrm{y}(\mathrm{x})=\mathrm{y}_{\mathrm{h}}(\mathrm{x})+\mathrm{y}_{\mathrm{p}}(\mathrm{x})$

<h2 class="content_sub_heading">Typen von DGLs</h2>
Triviale Differentialgleichung $$y^{(n)}=f(x)$$ Getrennte Variablen $$y^{\prime}=f(x) \cdot g(y)$$ Linear 1. Ordnung $$y^{\prime}+a(x) \cdot y=b(x)$$ Euler-homogen $$y^{\prime}=f\left(\frac{y}{x}\right)$$ Lineare Transformation $$y^{\prime}=f(a x+b y+c)$$ Gebrochen rationale Transformation $$y^{\prime}=f\left(\frac{a x+b y+c}{d x+e y+f}\right)$$ Linear höherer Ordnung mit konstanten Koeffizienten $$y^{(n)}+a_{n-1} y^{(n-1)}+\ldots+a_{1} y^{\prime}+a_{0} y=b(x)$$ Bernoulli $$y^{\prime}+g(x) \cdot y+h(x) \cdot y^{\alpha}=0$$ Ricatti $$y^{\prime}+g(x) \cdot y+h(x) \cdot y^{2}=k(x)$$

<h2 class="content_sub_heading">Form der Riccatischen DGL</h2>
\begin{align*}\quad y^{\prime}+g(x) \cdot y+h(x) \cdot y^{2}=k(x)\end{align*}

<h2 class="content_sub_heading">Form der Riccatischen DGL</h2>
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<h2 class="content_sub_heading">Allgemeine Lösung einer Riccatischen DGL bestimmen </h2>
Mit gegebener spezieller Lösung $y_{s}(x)$, kannst du die allgemeine Lösung folgendermaßen bestimmen:
<ol>
<li>Setze die Funktion $y(x)=v(x)+y_{s}(x)$ in deine Riccatische DGL ein. Nach einigem Umformen* erhältst du dann:  $\quad v^{\prime}+\left(g(x)+2 h(x) y_{s}(x)\right) \cdot v+h(x) \cdot v^{2}=0$ </li>
<li>Jetzt hast du eine Bernoullische DGL für $v(x)$ und $\alpha=2$. </li>
<li>Ähnlich wie beim Lösen der Bernoulli-DGL transformierst du nun deine DGL mit $v=z^{-1}=\frac{1}{z}$ und erhältst dadurch eine lineare DGL 1. Ordnung für $z$: \begin{align*}\quad z^{\prime}-\left(g(x)+2 h(x) y_{s}(x)\right) \cdot z-h(x)=0\end{align*} </li>
<li>Löse jetzt die lineare DGL 1. Ordnung. Dadurch erhältst du eine Lösung für $z$. </li>
<li>Jetzt setzt du diese allgemeine Lösung für $z(x)$ in die Transformationsgleichungen ein und transformierst zurück: \begin{align*}\quad y(x) &amp;=v(x)+y_{s}(x) =\frac{1}{z(x)}+y_{s}(x) \end{align*}</li>
</ol>

<h2 class="content_sub_heading">Differentialgleichung</h2>
Eine Differentialgleichung (DGL) ist eine Gleichung, die eine Funktion und ihre Ableitungen enthält. Die Lösung einer DGL ist also eine differenzierbare Funktion, die diese Gleichung erfüllt.

<h2 class="content_sub_heading">Kategorien von DGLs</h2>
gewöhnliche DGL: nur Ableitungen von einer Variablen
Beispiel: $y(x) = 4y^{\prime}(x) + y^{\prime \prime}(x)$
 
partielle DGL: mehrere Variablen mit partiellen Ableitungen
Beispiel: $y(x,t) = 4 \frac{\partial^2{y}}{\partial{t^2}} + 3 \frac{\partial{y}}{\partial{x}}$
 
Ordnung: Anzahl der höchsten Ableitung
Beispiel: $y = 4y^{\prime} + y^{\prime \prime}$ (diese DGL ist 2. Ordnung).
 
lineare DGL: nur Linearkombinationen der Funktion und ihrer Ableitungen
Beispiel: $y = 4x^2y^{\prime} + 3xy^{\prime \prime}$
 
nicht-lineare DGL: gesuchte Funktion hat Potenzen oder ist in anderen Funktionen verkettet
Beispiel: $y^{\prime} = sin(y)$
 
homogene DGL: es gibt keinen Term ohne $y$ (die gesuchte Funktion oder ihre Ableitungen)
Beispiel: $y + 2y^{\prime} + 3xy^{\prime \prime} = 0$
 
inhomogene DGL: es gibt Störfunktion (Term ohne $y$)
Beispiel: $y + 2y^{\prime} + 3xy^{\prime \prime} = 2x^2$
 
explizite DGL: höchste Ableitung steht alleine auf einer Seite. (Falls nicht, implizite DGL).

Aufgabenstellung:

Lösungsweg:

Lösung: