Question

The solution of the differential equaiton $x\frac{dy}{dx}+2y=x^2$ is _________.

Answer 1

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

$\Rightarrow \frac{dy}{dx}+\frac{2}{x}y=x$
Comparing it with linear differential equaiton of form:
$\frac{dy}{dx}+Py=Q$

$P=\frac{2}{x},Q=x$

So, the integrating factor is

$I.F.=e^{\int \frac{2}{x}dx}(\because I.F.=e^{\int pdx})$

$\Rightarrow I.F.=e^{2In|x|}$

$=e^{In|x{|}^2}$

$=|x{|}^2=x^2$

Therefore, the solution is

$y.x^2=\int x.x^{2}dx+c$

$(\because y(I.F.)=\int Q(I.F)dx+c)$

$\Rightarrow y.x^2=\int x^{3}dx+c$

$\Rightarrow y.x^2=\frac{x^4}{4}+c$

$\Rightarrow y=\frac{x^2}{4}+c{x}^{-2}$

Hence, the required solution is $y=\frac{x^2}{4}+c{x}^{-2}$