Question

For the given differential equation find the general solution.

$\frac{dy}{dx}+\frac{y}{x}=x^2$

Answer 1

Given, $\frac{dy}{dx}+\frac{y}{x}=x^2$
On comparing with the form $\frac{dy}{dx}+Py=Q,$ we get $P=\frac{1}{x}andQ=x^2$
$\therefore IF=e^{\int Pdx}=e^{\frac{1}{x}dx}\Rightarrow IF=e^{log|x|}=x$ $\because e^{logx}=x$
The solution of the given differential equation is given by
$y.IF=\int Q\times IFdx+C\Rightarrow yx=\int x^{3}dx+C\Rightarrow yx=\frac{x^4}{4}+C$