1.Show that the given equation is not exact but becomes exact when multiplied by the given integrating factor.Then solve the equation.
$$
x^2y^3+x(1+y^2)y^{\prime}=0,\qquad\mu(x,y)=1/xy^3.
$$
$$
\begin{aligned}
M&=x^2y^3& N=x+xy^2\\
My&=3x^2y^2& Nx=1+y^2 \text{not exact.}
\end{aligned}
$$
$$\left. \begin{array}{l}{ \displaystyle\frac { x ^ { 2 } y ^ { 3 } } { x y ^ { 3 } } + ( \frac { x } { x y ^ { 3 } } + \frac { x y ^ { 2 } } { x y ^ { 3 } } ) y ^ { \prime } = 0 }\\{ \displaystyle x + ( \frac { 1 } { y ^ { 3 } } + \frac { 1 } { y } ) y ^ { \prime } = 0 }\\{ M_y = N_x = 0 }\end{array} \right.$$
Hence$$\begin{aligned}
\exists \phi_{(x,y)} s.t. \phi _ { x } &= M \\ \phi &= \int x d x \\ &= \frac { 1 } { 2 } x ^ { 2 } + h ( y ) \\ \Phi _ { y } &= \frac { 1 } { y ^ { 3 } } + \frac { 1 } { y } \\ h ^ { \prime } ( y ) &= - \frac { 1 } { 2 } y ^ { - 2 } + \ln | y | + c \\\phi_{(x,y)}&=\frac{1}{2}x^2-\frac{1}{2}y^{-2}+\ln | y |+c
\end{aligned}
$$
