Question

Resolver a Equação diferencial homogênea utilizando o método da substituição:

Answer 1

$\large\boxed{\begin{array}{l}\rm\dfrac{dy}{dx}=\dfrac{x-y}{x}\\\\\rm\dfrac{dy}{dx}=1-\dfrac{y}{x}\\\\\rm\dfrac{dy}{dx}+\dfrac{1}{x}y=1\\\\\rm e^{\displaystyle\rm\int\dfrac{1}{x}\,dx}=e^{\ln x}=x\\\\\rm \mu(x)\dfrac{dy}{dx}+\mu(x)\cdot\dfrac{1}{x}\cdot y=\mu(x)\cdot 1\\\\\rm x\dfrac{dy}{dx}+\backslash\!\!\!x\cdot\dfrac{1}{\backslash\!\!\!x}\cdot y=x\\\\\rm\dfrac{d}{dx}(y\cdot x)=x\\\rm d(y\cdot x)=x\,dx\\\displaystyle\rm\int d(y\cdot x)=\int x\,dx\\\rm y\cdot x=\dfrac{1}{2}x^2+k\end{array}}$

$\large\boxed{\begin{array}{l}\rm y=\dfrac{1}{\diagup\!\!\!x}\cdot\dfrac{1}{2}\diagup\!\!\!\!x^2+\dfrac{1}{x}\cdot k\\\\\rm y(x)=\dfrac{x}{2}+\dfrac{k}{x}\end{array}}$