Exact Differential Equations $(axy^2+by)dx+(bx^2y+ax)dy=0$. $M(x,y)dx + N(x,y)dy=0$ is said to be a perfect differential when 
$$\frac{\partial (M(x,y))}{\partial y}=\frac{\partial (N(x,y))}{\partial x}$$
Let $M_y=\frac{\partial (M(x,y))}{\partial y}$ and $N_x=\frac{\partial (N(x,y))}{\partial x}$.
In case if:
$\frac{M_y-N_x}{N(x,y)}$ is only a function of x only (say $f(x)$) then it has a integrating factor $e^{\int f(x)dx}$.
But if $\frac{M_y-N_x}{N(x,y)}$ is a constant then what will be the integrating factor?

Say for this problem:
  $(axy^2+by)dx+(bx^2y+ax)dy=0$

Is there any other method to solve this differential equation?

If $a\neq0$ and $b\neq0$ then integrating factor of equation
$$(axy^2+by)dx+(bx^2y+ax)dy=0$$
is
$$\\\frac{1}{{{x}^{\frac{2 b+a}{b+a}}} {{y}^{\frac{b+2 a}{b+a}}}}$$
and solution
$$\\\frac{x^2y^2-xy}{{{x}^{\frac{2 b+a}{b+a}}} {{y}^{\frac{b+2 a}{b+a}}}}=C$$
