Answer to Question #204122 in Differential Equations for rikajk

Question

HOMOGENEOUS EQUATION

Solve the given equation below. Show complete solution.

xy' = x + y

Answer 1

Ans:-

$xy'=x+y$

$\dfrac{dy}{dx}=\dfrac{x+y}{x}\ \ \ \ \ \ \ \ \ \ \ ...1$

Solving $\dfrac{dy}{dx}$ by putting $\ \ y=vx$

differentiating w. r. t. x

$\dfrac{dy}{dx}=x\dfrac{dv}{dx}+v\dfrac{dx}{dx}$

$\dfrac{dy}{dx}=x\dfrac{dv}{dx}+v$

Putting value of $\dfrac{dy}{dx}$ and y=vx in (1)

$\dfrac{dy}{dx}=\dfrac{x+y}{x}$

$x\dfrac{dv}{dx}+v=\dfrac{x+vx}{x}\\$

$x\dfrac{dv}{dx}+v=1+v\\$

$\dfrac{dv}{dx}=\dfrac{1}{x}$

Integrating both sides

$\int dv=\int \dfrac{dx}{x}$

$v=log|x| +c$

putting $v=\dfrac{y}{x}$

$\dfrac{y}{x}=log|x| +c$

$y=xlog|x|+cx$