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

Answer 1

Answer: (D)

We have, $x=\frac{dy}{dx}+2y=x^2$
$\Rightarrow \frac{dy}{dx}+\frac{2}{x}y=x$
The above equation is a linear differential equation in y.
$\therefore IF=e^{\int \frac{2}{x}dx}=e^{2logx}=x^2$
Hence, required solution will be
$y.x^2=\int x.x^{2}dx+C_1$
$\Rightarrow y{x}^2=\frac{x^4}{4}+C_1$
$\Rightarrow y{x}^2=\frac{x^4+4C_1}{4}$
$\Rightarrow y=\frac{x^4+C}{4x^2}$ $[\because 4C_1=C]$