Q. The solution of $\left( 1 + x^{2} \right)\frac{dy}{dx} + 2xy - 4x^{2} = 0$ is
A) $3x\left( 1 + y^{2} \right) = 4y^{3} + c$
B) $3y\left( 1 + x^{2} \right) = 4x^{3} + c$
C) $3x\left( 1 - y^{2} \right) = 4y^{3} + c$
D) $3y\left( 1 + y^{2} \right) = 4x^{3} + c$
Explanation
(b)
Given, $\frac{dy}{dx} + \frac{2x}{1 + x^{2}}.y = \frac{4x^{2}}{1 + x^{2}}$
$\therefore$ $IF = e^{\int_{}^{}{\frac{2x}{1 + x^{2}}\ dx}} = e^{\log\left( 1 + x^{2} \right) = (1 + x^{2})}$
$\therefore$ Complete solution is
$y.\left( 1 + x^{2} \right) = \int_{}^{}{(1 + x^{2})}.\ \frac{4x^{2}}{1 + x^{2}}\ dx$
$\Rightarrow$ $y\left( 1 + x^{2} \right) = \frac{4x^{3}}{3} + c_{1}$
$\Rightarrow$ $3y\left( 1 + x^{2} \right) = 4x^{3} + c$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.