Given a function,
$$u_{xy}=x^2y ,u(x,y) $$
I was wondering if there was a better way to solve this problem aside from using basic calculus, particularly using differential equations
Obviously I can do $$F(x)F(y) + c(y) + c(x)$$ by integrating but could I use a separation of variables for $u_{xy}$ or some other method
You can see that $f(x)+g(y)$ is the solution of $u_{xy}=0$.
For the term $x^2y$ on the right hand side; we have
$\frac{1}{DD'}x^2y=\frac{1}{D}x^2(y^2/2)=(x^3/3)(y^2/2)=x^3y^2/6$.
Thus the solution of given pde is $u(x,y)=f(x)+g(y)+\frac{x^3y^2}{6}$