Question

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

• A. $y=cx^{-3}-\frac{x^{2}}{4}$
• B. $y=cx^{3} - \frac{x^2}{4}$
• C. $y=cx^{2} + \frac{x^3}{5}$
• D. $y=cx^{-2} + \frac{x^3}{5}$

Answer 1

Answer: (D)

$y=cx^{-2} + \frac{x^3}{5}$

Answer 2

Given differential equation is $\frac{dy}{dx}+\frac{2}{x}.y=x^{2}$ This is of the linear form. $\therefore P=\frac{2}{x}, Q=x^{2}$ $I.F.=e^{\int \frac{2}{x}dx}=e^{log\,x^2}=x^{2}$ Solution is $yx^{2}=\int\,x^{2}\,.x^{2}\,dx+c=\frac{x^{5}}{5}+c$ $y=\frac{x^{3}}{5}+cx^{-2}$