Answer
\[y(x)=C(1+x)e^x\]
Work Step by Step
$(1+x)y'=y(2+x)$ ____(1)
(1) is Separable differentiable equation
\[y'=\frac{dy}{dx}=\frac{y(2+x)}{1+x}\]
Separating variable,
\[\frac{dy}{y}=\left(\frac{2+x}{1+x}\right)dx\]
Integrating,
\[\int\frac{dy}{y}=\int\left(\frac{2+x}{1+x}\right)dx+C_{1}\]
$C_{1}$ is constant of integration
\[\ln|y|=\int\left[1+\frac{1}{1+x}\right]dx+C_{1}\]
\[\ln|y|=x+\ln|1+x|+C_{1}\]
\[y=e^{x+\ln |1+x|+C_{1}}\]
\[y(x)=C(1+x)e^x\]
Where $C=e^{C_1}$
Hence general solution of (1) is $y(x)=C(1+x)e^x$
