Question

ALGUEM AJUDA POR FAVOR?
Determinar a equação diferencial linear

Answer 1

$\Large\boxed{\begin{array}{l}\underline{\rm Equac_{\!\!,}\tilde oes~diferenciais~lineares~de~1^a~ordem}\\\sf D~\!\!efinic_{\!\!,}\tilde ao:\dfrac{dy}{dx}+p(x)\cdot y=q(x)\\\sf fator~integrante:\mu(x)=e^{\displaystyle\sf\int p(x)dx}\end{array}}$

$\Large\boxed{\begin{array}{l}\sf\dfrac{dy}{dx}+\bigg(3+\dfrac{1}{x}\bigg)y=\dfrac{e^{-3x}}{x}\\\underline{\rm c\acute alculo~do~fator~integrante:}\\\sf\mu(x)=e^{\displaystyle\sf\int\bigg(3+\dfrac{1}{x}\bigg)dx}\\\\\sf \mu(x)=e^{3x+\ell n(x)}=e^{3x}\cdot e^{\ell nx}\\\sf \mu(x)=xe^{3x}\end{array}}$

$\Large\boxed{\begin{array}{l}\underline{\rm multiplicando~a~equac_{\!\!,}\tilde ao~diferencial}\\\underline{\rm~pelo~fator~integrante}\\\sf xe^{3x}\dfrac{dy}{dx}+xe^{3x}\bigg(3+\dfrac{1}{x}\bigg)y=xe^{3x}\cdot\dfrac{e^{-3x}}{x}\\\sf\dfrac{d}{dx}[e^{3x}\cdot xy]=1\\\sf d[e^{3x}\cdot xy]=dx\\\underline{\rm integrando~dos~dois~lados~temos\!:}\\\displaystyle\sf\int d[e^{3x}\cdot xy]=\int dx\\\sf e^{3x}\cdot xy=x+k\\\underline{\rm isolando~y~temos:}\\\sf y=\dfrac{x}{xe^{3x}}+\dfrac{k}{xe^{3x}}\end{array}}$

$\Large\boxed{\begin{array}{l}\underline{\rm simplificando~temos:}\\\huge\boxed{\boxed{\boxed{\boxed{\sf y=\dfrac{1}{e^{3x}}+\dfrac{k}{xe^{3x}}}}}}\end{array}}$