Question

The general solution of the differential equation $xy(y+1)dy=(1+x^2)dx$ is
(where $c$ is constant of integration)

• A. $y+\frac{y^2}{2}=\ln |x|+\frac{x^2}{2}+c$
• B. $\frac{y}{2}+\frac{y^2}{3}=2\ln |x|+\frac{x^2}{2}+c$
• C. $\frac{y^2}{2}+\frac{y^3}{3}=\ln |x|+x+c$
• D. $\frac{y^2}{2}+\frac{y^3}{3}=\ln |x|+\frac{x^2}{2}+c$

Answer 1

The correct option is D $\frac{y^2}{2}+\frac{y^3}{3}=\ln |x|+\frac{x^2}{2}+c$

Given : $xy(y+1)dy=(1+x^2)dx$
$\Rightarrow y(y+1)dy=(\frac{1}{x}+x)dx$
Integrating both sides, we get
$\frac{y^2}{2}+\frac{y^3}{3}=\ln |x|+\frac{x^2}{2}+c$