# Partial differential equations question

I'm really struggling to do the following question I know Separation of variables will come into it but I'm unsure of how to separate these particular variable. I am also struggling with when to substitute the u_{x} back in.

THE QUESTION:
Consider the equation u_{xx} u_{xy} = 0.
(a) Find the general solution. (Hint: Substitute v = u_{x}.)
Original post by jmcuni
I'm really struggling to do the following question I know Separation of variables will come into it but I'm unsure of how to separate these particular variable. I am also struggling with when to substitute the u_{x} back in.

THE QUESTION:
Consider the equation u_{xx} u_{xy} = 0.
(a) Find the general solution. (Hint: Substitute v = u_{x}.)

I presume they want you solve
v_x - v_y = 0
so v could be pretty much any differentiable function with a fairly obvious argument in terms of x and y. It would probably help to see the full questtion to see where its going.
Original post by mqb2766
I presume they want you solve
v_x - v_y = 0
so v could be pretty much any differentiable function with a fairly obvious argument in terms of x and y. It would probably help to see the full questtion to see where its going.

The whole question is:
Consider the equation u_{xx} u_{xy} = 0.
(a) Find the general solution. (Hint: Substitute v = u_{x}.)
(b) Find the most general solution of this equation which satisfies the conditions
u(x,0) = e^{−x} and u_{y}(x,0) = −e^{−x}.
Is it unique?
Original post by jmcuni
The whole question is:
Consider the equation u_{xx} u_{xy} = 0.
(a) Find the general solution. (Hint: Substitute v = u_{x}.)
(b) Find the most general solution of this equation which satisfies the conditions
u(x,0) = e^{−x} and u_{y}(x,0) = −e^{−x}.
Is it unique?

For a, did you get the simple characteristic lines for v ...?
Original post by jmcuni
The whole question is:
Consider the equation u_{xx} u_{xy} = 0.
(a) Find the general solution. (Hint: Substitute v = u_{x}.)
(b) Find the most general solution of this equation which satisfies the conditions
u(x,0) = e^{−x} and u_{y}(x,0) = −e^{−x}.
Is it unique?

For a, did you get the simple characteristic lines for v ...?