Solving a PDE for z(x,y) with Total and 2nd Derivatives: Step-by-Step Guide

[chacal10]

TL;DR

I have not been able to solve this PDE; need some help pls

as you can see, z(x,y) is a function of x, y; and y is a function of x, therefore y'(x) is the total derivative of "y" respect to "x", and y"(x) is the 2nd derivative. y'(x)^2 is just the square of the derivative of y respect to x

I don't have boundary or initial conditions, so you can make up any if that simplifies finding the solution

Thank you !

 

[jedishrfu]

Have you tried the separation of variables strategy?

Assume $Z(x,y) = X(x)Y(y)$

 

[jedishrfu]

Here's a small tutorial on the technique:

http://tutorial.math.lamar.edu/Classes/DE/SeparationofVariables.aspx

and some lecture notes on the topic:

http://pi.math.cornell.edu/~rand/randdocs/PDE_handout/PDE15.pdf

 

[chacal10]

Let me try that. I think the problem with separation of variables is the total derivative terms y'(x) and y"(x) that multiply the partial derivatives

 

[TheDS1337]

I don't know much about PDEs but I see you're looking for two functions here, shouldn't there be a system of coupled equations instead of just one?

 

[chacal10]

@TheDS1337
Is actually one function Z(x,y), but "y" itself is a function of "x" (unknown at this time)
What I am trying to find as the "solution" to this PDE is Z(x,y), i.e. something like "...Z(x,y) = A*cos(y-wx) + exp(-y*x)..." or something like that.
Note that the first 2 parts of the PDE look suspiciously close to a wave equation (if you replace "x" by "t"), but then the 3rd part messes the whole thing up

Thank you all!