$$
(xy^{2}+bx^{2}y)+(x+y)x^2y^{\prime}=0
$$
$$
\begin{aligned}
M&=xy^2+bx^2y\qquad N=(x+y)x^{2}=x^3+yx^2\\
M_y&=2yx+bx^2\qquad N_x=3x^2+2yx\\
\therefore &b=3
\end{aligned}
$$
$$
(xy^{2}+3x^{2}y)+(x+y)x^2y^{\prime}=0
$$
$$
\begin{aligned}
M&=xy^2+3x^2y\qquad N=(x+y)x^{2}=x^3+yx^2\\
M_y&=2yx+3x^2\qquad N_x=3x^2+2yx\\
\therefore &M_y=N_x\quad\therefore \text{the function is exact.}
\end{aligned}
$$
$$
\begin{aligned}
\exists ~\text{s.t.} \varphi_x&=M\\
\varphi&=\int M d x\\
&=\int xy^2+3x^2ydx\\
&=\frac{x^2y^2}{2}+x^3y+h(y)\\
\varphi_y&=\frac{2yx^2}{2}+x^3+h^{\prime}(y)\\
&=x^2y+x^3=h^{\prime}(y)\\
&=x^3+x^2y\\
h^{\prime}(y)&=0\\
h(y)&=\text{constant}\\
\varphi&=\frac{x^2y^2}{2}+x^3y=c
\end{aligned}
$$
