Prove that Heaviside's function H(x) is a solution for equation
$$ \frac{\partial^2 u}{\partial x_1\partial x_2}=0 $$ $$$$ $$$$ For example, to solve that function
$$ F(x)=e^{2x}H(x)$$
is solution for equation $$ \frac{\partial u}{\partial x}-2=0 $$
I should show
$$ F'(x)-2F(x)=\delta_0$$
But how should I prove the first example, where I have mixed derivative?
Thank you in advance
$$$$$$$$ Edit:
I've started:
$$ [H(x_1,x_2)]''[\Phi]=\int\limits_{-\infty}^{\infty}\int\limits_{-\infty}^{\infty} H(x_1,x_2)\Phi(x_1,x_2)dx_1dx_2=\int\limits_{-\infty}^{\infty}\int\limits_{0}^{\infty} \frac{\partial}{\partial x_1}\Phi(x_1,x_2)dx_1dx_2=\int\limits_{-\infty}^{\infty}\Phi(0,x_2)dx_2$$
But here I've stoped, what next?