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

The Correct Option is 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. $\therefore IF=e^{\int \frac{2}{x} dx}\,=e^{2\,log\,x}=x^{2}$ Hence, required solution will be $ y . x^{2}=\int x . x^{2}\,dx+C_{1}$ $\Rightarrow yx^{2}=\frac{x^{4}}{4}+C_{1}$ $\Rightarrow yx^{2}=\frac{x^{4}+4C_{1}}{4}$ $\Rightarrow y=\frac{x^{4}+C}{4x^{2}}$ $[\because 4C_{1}=C]$