Question

A EDO dy/dx =3-2y, tem como fator integrante I=e^2x. Assinale a alternativa correta que contém a sua solução geral.

Answer 1

$\frac{dy}{dx}=3-2y\ \to\ dy.\frac{1}{3-2y}=dx\\ \int \frac{1}{-2y+3}\ dy=\int 1\ dx\\\\ Resolvendo\ \int \frac{1}{-2y+3}\ dy:\\\\u=-2y+3\\ \frac{du}{dy}=-2\ \to\ \frac{-du}{2}=dy\\\\ \int \frac{1}{u}.\frac{-1}{2}\ du=\frac{-1}{2}\int \frac{1}{u}\ du=\frac{-1}{2}.ln(u)\\ \int\frac{1}{-2y+3}\ dy=\frac{-ln(-2y+3)}{2}+c_1\\\\ Logo:\\\\ \frac{-ln(-2y+3)}{2}+c_1=x+c_2\ \to\ -ln(-2y+3)=2x+2c_1\\ ln(-2y+3)=-2x-2c_1\ \to\ ln(-2y+3)=ln(e^{-2x-2c_1})\\\\ -2y+3=e^{-2x-2c_1}\ \to\ \| \ y=\frac{-(e^{-2x-2c_1}-3)}{2}\ \|$

Answer 2

Resposta:

y= 3/2 + Ce^-2x

Explicação passo-a-passo: