Simple linear first order PDE

Question

I did not take course in PDEs yet, but I need to solve this equation for physics. I know the solution but I'm not sure how exactly should I solve it.

$$-A f(x, y) = \frac{\partial f(x, y)}{\partial x}$$

Where A is constant and boundary condition is:

$$f(x, x) = f(y, y) = 1$$

Thx in advance.

Answer 1

I'll give you a hint: try to solve this problem by setting f(x,y) to be equal to the multiplication of functions $\alpha(x)$ and $\beta(y)$. Try to find out what $\alpha(x)$ is. Then apply the condition you specified. See how far you can go!

Answer 2

Integrating with respect to $x$ the differential equation $\partial_x f(x,y) = -A f(x,y)$, we get $f(x,y) = e^{-Ax}f (0,y)$. Indeed, the problem is similar to the ODE $f' = -Af$, but here, the "integration constant" depends on $y$. Now, since $f(y,y) = 1$, we are left with $f (0,y)=e^{Ay}$, and finally $f(x,y) = e^{A(y-x)}$.