TITLE: Solving differential equation with two variables
QUESTION [7 upvotes]: How to solve a differential equation of the following form:
$\dfrac{ax}{\dfrac{\partial{f(x,y)}}{\partial x}} + \dfrac{by}{\dfrac{\partial{f(x,y)}}{\partial y}}=-1 $
Where, $f(x_0,y_0)=c$.

REPLY [2 votes]: $$\dfrac{ax}{\dfrac{\partial{f(x,y)}}{\partial x}} + \dfrac{by}{\dfrac{\partial{f(x,y)}}{\partial y}}=-1 \quad \text{ with condition} \quad f(x_0,y_0)=c$$
This is a non-linear PDE : The given condition isn't sufficient to determine a unique solution.
For example, the method of separation of variables leads to :
$$f(x,y)=\frac{a(x^2-x_0^2)}{2\lambda}-\frac{b(y^2-y_0^2)}{2(\lambda+1)}+c$$
which is a family of solutions of the PDE fitting with the given condition.
This is true any constant $\lambda$, except $0$ and $-1$. So, they are an infinity of solutions.
Note : The solution $\frac a2(x^2-x_0^2)-\frac b{\color{red}{4}}(y^2-y_0^2)+c$ cited by "polfosol" is the particular solution in case of $\lambda=1$.