Answer
\[y(x)=\frac{1}{1-Ce^{\frac{x^3}{3}}}\]
Work Step by Step
Type:- Separable Differential Equation
\[\frac{dy}{dx}=x^2y(y-1)\]
Separating variables,
\[\frac{dy}{y(y-1)}=x^2dx\]
Integrating,
\[\int\frac{1}{y(y-1)}dy=\int x^2dx+C_{1}\]
$C_{1}$ is constant of integration
\[\int\left[\frac{1}{y-1}-\frac{1}{y}\right]dy=\frac{x^3}{3}+C_{1}\]
\[\ln|y-1|-\ln|y|=\frac{x^3}{3}+C_{1}\]
\[\ln\left|\frac{y-1}{y}\right|=\frac{x^3}{3}+C_{1}\]
\[\ln\left|1-\frac{1}{y}\right|=\frac{x^3}{3}+C_{1}\]
\[1-\frac{1}{y}=Ce^{\frac{x^3}{3}}\]
Where $C=e^{C_{1}}$
\[1-Ce^{\frac{x^3}{3}}=\frac{1}{y}\]
\[y(x)=\frac{1}{1-Ce^{\frac{x^3}{3}}}\]
Hence general solution is $y(x)=\Large\frac{1}{1-Ce^{\frac{x^3}{3}}}$.
