Question: show that the given equation is not exact but becomes exact when multiplied by the given integrating factor.Then solve the equation.
The equation is:
\begin{equation*}
x^2y^3+x(1+y^2)y'=0
\end{equation*}
The integrating factor $\mu(x,y)=\frac{1}{xy^3}$
Solution:
First,verify that the initial equation is not exact:
\begin{align*}
M(x,y) &=x^2y^3 & N(x,y) &=x+xy^2\\
M_y &=3x^2y^2 & N_x &=1+y^2\\
\end{align*}
$M_y \neq N_x$,the equation is not exact.
Next, multiplying $\mu$ on both sides,
\begin{align*}
\mu x^2y^3+\mu x(1+y^2)y'=0\\
x+\frac{1+y^2}{y^3}y'=0
\end{align*}
Denote $P(x,y)=x$ and $Q(x,y)=\frac{1+y^2}{y^3}$
Check exactness:
\begin{equation*}
P_y=Q_x=0
\end{equation*}
The new equation is exact.
Now solve the equation:
\begin{align*}
\frac{\partial\psi}{\partial x}=x\\
\psi=\frac{1}{2}x^2+g(y)\\
\frac{\partial\psi}{\partial y}=g'(y)=y^{-3}+y^{-1}\\
g(y)=-\frac{1}{2}y^{-2}+ln|y|
\end{align*}
So the solution to the equation is
\begin{equation*}
\psi(x,y)=\frac{x^2}{2}-\frac{1}{2y^2}+ln|y|=c
\end{equation*}
for $c$ being constant.
