Q. Solution of the differential equation $\frac{dy}{dx} + \frac{y}{x} = x^{2}$ is
A) $y = \frac{x^{2}}{4} + cx^{- 2}$
B) $y = x^{- 1} + cx^{- 3}$
C) $y = \frac{x^{3}}{4} + cx^{- 1}$
D) $xy = x^{2} + c$
Explanation
(c)
Given, $\frac{dy}{dx} + \frac{y}{x} = x^{2}$
$\therefore$ $IF = e^{\int_{}^{}{\frac{1}{x}\ dx}} = e^{\log x} = x$
$\therefore$ Complete solution is
$y.x = \int_{}^{}{x.x^{2}}dx + c$
$\Rightarrow$ $y.x = \frac{1}{4}\ x^{4} + c$
$\Rightarrow$ $y = \frac{1}{4}\ x^{3} + cx^{- 1}$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.