I hope that somebody may help me with this partial differential equation
(d^2/(dxdy)) f=0,
I do appreciate if the solution is step by step. Cheers.
$f(x,y)=a_1x+a_2y+a_3$
Take second derivative, one with respect to $x$ and one with respect to $y$. You will get zero. There were no steps.
Method of characteristics? I would just integrate. Integrating both sides of $\displaystyle \frac{\partial f}{\partial x\partial y}= 0$, with respect to x, gives $\displaystyle \frac{\partial f}{\partial y}= g(y)$. That is, integrating 0 with respect to x, the integrand is a "constant" but, since the derivative was with respect to x only, that "constant" could be a function of y since, in taking the partial derivative with respect to x, y is treated as a constant. The derivative, with respect to x, of any function of y alone is 0,
From $\displaystyle \frac{\partial f}{\partial y}= g(y)$, integrating with respect to y gives $\displaystyle f(x,y)= G(y)+ H(x)$ where G(x) an anti-derivative of g(y) and H(x) is the "constant of integration" when integrating with respect to y. Again, since the differentiation we are "undoing" is with respect to y, any function of x only is treated as a constant.
That is, the general solution to $\displaystyle \frac{\partial^2 f}{\partial x\partial y}= 0$ is $\displaystyle f(x, y)= G(y)+ H(x)$ where G(y) can be any differentiable function of y and H(x) any differentiable function of x.
(The solution SlipEternal gives assumes G and H are linear which is not necessarily true.)