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

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

Divide each sides by $x$

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

Integrating factor $=e^{\int \frac{2}{x}dx}=e^{2\ln x}=e^{\ln x^2}=x^2$

Hence solution is given by,

$y\left(x^2\right)=\int x^2\cdot xdx+c$

$\Rightarrow x^{2}y=\int x^{3}dx+C=\frac{x^4}{4}+c$

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