# Second order hyperbolic PDE

Can anyone help me with finding general solutions of this pde with given initial conditions.

$$\frac {\partial^2}{\partial x \partial y} f(x, y) = \alpha f(x, y), \quad x>0$$

$$f(x,0)=0$$ $$f(0,y)=a(y) = \begin{cases} 0, & y < 10 \\ 1, & y \ge 10 \end{cases}$$

$a(y)$ is not continuous.

• What have you tried? Also the second derivative of an unknown function equals the functions itself times a constant. Does this give you any hints? – Rumplestillskin Jun 30 '18 at 2:44
• I have tried using separation of variables but I don't think it gives a correct answer because a(y) is not continuous.... Tbh I'm not expert in PDE – MDAN Jun 30 '18 at 2:51
• Separation of variables? Can you write down your attempt under the question. – Rumplestillskin Jun 30 '18 at 4:04
• math.stackexchange.com/questions/2269063 – doraemonpaul Jun 30 '18 at 12:34

The general solution is $f(x,y)=\int_0^xF(s)I_0\left(\dfrac{2\sqrt{y(x-s)}}{\sqrt\alpha}\right)ds+\int_0^yG(s)I_0\left(\dfrac{2\sqrt{x(y-s)}}{\sqrt\alpha}\right)ds$
$f(x,0)=0$ :
$\int_0^xF(s)~ds=0$
$F(s)=0$
$\therefore f(x,y)=\int_0^yG(s)I_0\left(\dfrac{2\sqrt{x(y-s)}}{\sqrt\alpha}\right)ds$