Find the solution of the equation uxy = ux

[kasse]

Homework Statement

Find the solution of the equation

u_xy = u_x

The Attempt at a Solution

u_x = p

p_y = p

ln p = y + c*

p = ce^y

But them, what is u?

 

[HallsofIvy]

Well, how would you solve the ORDINARY differential equation. dp/dx= p? That's seems to me very easy. Warning: because this is NOT an ordinary differential equation, the "constant" in the solution may be a function of y. Also remember that the general solution to a partial differential equation involves unknown FUNCTIONS.

 

[kasse]

Just integrate wrt. x so that

u(x,y) = e^y Int[c(x)]dx + g(y) where c(x) and g(y) are arbitrary?