Solutions of the wave equation's little brother

[quasar987]

Suppose I showed that a function f(z,t) of which I do not know the form explicitely satisfies the following pde:

$$\frac{\partial f}{\partial z}=-\frac{1}{v}\frac{\partial f}{\partial t}$$

While it is certain that functions of the type g(z-vt) are solutions to the pde, does it mean that my f(z,t) is of this form necessarily?

 

[Astronuc]

Well, one could try a general solution

$$f(z,-vt)\,=\,g(z)h(-vt)$$ and see where that takes one.

 

[quasar987]

One understand that general solutions to pde are quite different from general solution to ode and one has no experience in dealing with the former.

 

[HallsofIvy]

Since that is a linear equation, solutions may also be any sum of functions of that type.

 

[LeonhardEuler]

Since that is a linear equation, solutions may also be any sum of functions of that type.

But sums of functions of that type are also functions of that type.