Resolva a equação diferencial. x dy/dx - 4y = x

Passo 1

Vamos resolver a equação diferencial . Essa é uma equação diferencial de primeira ordem e linear. Para resolver, vamos usar o método do fator integrante, que é uma técnica comum para esse tipo de equação.

Passo 2

Primeiro, vamos deixar a equação no formato padrão de uma equação diferencial linear: .
Dividindo ambos os lados da equação original por (considerando que ):
Aqui, identificamos que e .

Passo 3

Agora, precisamos encontrar o fator integrante , que é dado por:
Substituindo :
Como estamos trabalhando com , podemos simplificar para:

Passo 4

Multiplicamos toda a equação diferencial pelo fator integrante :
Simplificando:

Passo 5

O lado esquerdo da equação agora é a derivada do produto :
Isso nos permite integrar ambos os lados em relação a :

Passo 6

Integrando ambos os lados:
Onde é a constante de integração.

Passo 7

Agora, resolvemos para :

Resposta

A solução geral da equação diferencial é:

Exercícios relacionados

<html><head></head><body><div>Resolva a equação diferencial. <span class="math-inline ">
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<html><head></head><body><div>Use coordenadas cilíndricas para encontrar o volume dos seguintes sólidos: O sólido no primeiro octante limitado pelo cilindro <span class="math-inline ">
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http://www.w3.org/2000/svg” width=“5.169ex” height=“1.692ex” role=“img” focusable=“false” viewBox=“0 -666 2284.6 748”><title>r = 1</title><g stroke=“currentColor” fill=“currentColor” stroke-width=“0” transform=“scale(1,-1)”><g data-mml-node=“math”><g data-mml-node=“mi”><path data-c=“1D45F” d=“M21 287Q22 290 23 295T28 317T38 348T53 381T73 411T99 433T132 442Q161 442 183 430T214 408T225 388Q227 382 228 382T236 389Q284 441 347 441H350Q398 441 422 400Q430 381 430 363Q430 333 417 315T391 292T366 288Q346 288 334 299T322 328Q322 376 378 392Q356 405 342 405Q286 405 239 331Q229 315 224 298T190 165Q156 25 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 114 189T154 366Q154 405 128 405Q107 405 92 377T68 316T57 280Q55 278 41 278H27Q21 284 21 287Z”></path></g><g data-mml-node=“mo” transform=“translate(728.8,0)”><path data-c=“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></g><g data-mml-node=“mn” transform=“translate(1784.6,0)”><path data-c=“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></g></g></g></svg></mjx-container></span> e pelo plano <span class="math-inline ">
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Calcule as seguintes integrais:
Usando integração por partes, calcule