The general solution of the differential equation $\frac{d y}{d x}+\frac{2}{x} y=x^2$ is
The general solution of the differential equation $\frac{d y}{d x}+\frac{2}{x} y=x^2$ is
$y=c x^{-3}-\frac{x^2}{4}$
$y=c x^3-\frac{x^2}{4}$
$y=c x^2+\frac{x^3}{5}$
$y=c x^{-2}+\frac{x^3}{5}$
Solution
Given differential equation is
$
\frac{d y}{d x}+\frac{2}{x} \cdot y=x^2
$
This is of the linear form.
$
\begin{aligned}
& \therefore P=\frac{2}{x}, Q=x^2 \\
& \text { I.F }=e^{\int \frac{2}{x} d x}=e^{\log x^2}=x^2
\end{aligned}
$
Solution is
$
\begin{aligned}
& y \cdot x^2=\int x^2 \cdot x^2 d x+c=\frac{x^5}{5}+c \\
& y=\frac{x^3}{5}+c x^{-2}
\end{aligned}
$