We try to solve the partial differential equations for the unknown 2-variables function $f$:
1) $\dfrac{\partial f}{\partial y}=0$
2) The system of equations $\dfrac{\partial^2 f}{\partial^2 y}=2xy$ and $\dfrac{\partial^2 f}{\partial x\partial y}=y^2+2$
My try:
1) The solution to the first equation is the function $f(x,y)=g(x)$ where $g$ is any function which depends only on $x$.
2) The solution to the first equation of the system is $$f(x,y)=\dfrac{1}{3}xy^3+c_1(x)y+c_2(x)$$ The solution to the second equation of the system is $$f(x,y)=\dfrac{1}{3}xy^3+2xy+k_1(y)+k_2(x)$$
Hence a common solution to the two equations would be
$$f(x,y)=\dfrac{1}{3}xy^3+2xy+c(x)$$
Is my try correct? thank you for your help !
