Question

Find the general solution of the differential equation $x\frac{dy}{dx}+2y=x^2(x\ne 0)$

Answer 1

$x\frac{dy}{dx}+2y=x^2$ $(x\ne 0)$

$\frac{dy}{dx}+\frac{2y}{x}=x$ $(x\ne 0)$

Integrating,

$e^{\int Pdx}$

$⟹e^{2\int \frac{l}{x}dx}$

$⟹e^{2lnx}$

$⟹x^2$

General solution,

$y\left(IF\right)=\int Q\left(IF\right)$

$y{x}^2=\int x\left(x^2\right)dx$

$x^{2}y=\frac{x^4}{4}+C$