Question

The solution of differential equation $x\frac{dy}{dx}+2y=x^2$ is

• A. $y=\frac{x^2+C}{4x^2}$
• B. $y=\frac{x^2}{4}+C$
• C. $y=\frac{x^4+C}{x^2}$
• D. $y=\frac{x^4+C}{4x^2}$

Answer 1

The correct option is D $y=\frac{x^4+C}{4x^2}$

$x\frac{dy}{dx}+2y=x^2$
$\Rightarrow \frac{dy}{dx}+\frac{2}{x}y=x$
$I.F.=\exp \left(\int \frac{2}{x}dx\right)=x^2$
$\therefore$ The solution is $y\cdot x^2=\int x\cdot x^{2}dx$
$\Rightarrow y\cdot x^2=\frac{x^4}{4}+C_1$

$\Rightarrow y\cdot x^2=\frac{x^4+4C_1}{4}$

$\Rightarrow y=\frac{x^4+C}{4x^2}\text{where}C=4C_1$ is a constant.