Question

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

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

Answer 1

The correct option is A $\frac{x^2}{4}+C{x}^{-2}$

$x\frac{dy}{dx}+2y=x^2\Rightarrow \frac{dy}{dx}+\frac{2y}{x}=x$
$I.F=e^{\int Pdx}=e^{\int \frac{2}{x}dx}=e^{2logx}=e^{log{x}^2}=x^2$
The solution of the differential equation is $y.x^2=\int x.x^{2}dx$
$y.x^2=\int x^{3}dxy.x^2=\frac{x^4}{4}+Cy=\frac{x^2}{4}+C{x}^{-2}$