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Solving partial differential equations

I was wondering if you could solve the PDE

z(x,y)_x - \frac{z(x,y)}{x} = 0

by just using the usual separation of variables technique to get

z = cx

, where

c

is a constant?

Also how would you solve say

z(x,y)_{yy} = 0

? Can you treat it as a total derivative and use the method for second order linear differential equations? that would give [text]z = c_1 + c_2y ?

Any help would be great!
Thanks

Reply 1

Smaug123

Original post by John taylor

I was wondering if you could solve the PDE

z(x,y)_x - \frac{z(x,y)}{x} = 0

by just using the usual separation of variables technique to get

z = cx

, where

c

is a constant?

Also how would you solve say

z(x,y)_{yy} = 0

? Can you treat it as a total derivative and use the method for second order linear differential equations? that would give [text]z = c_1 + c_2y

?

Any help would be great!
Thanks
Second question: yes. Because the problem is independent of x, you can pretty much ignore it until the end, when you have to remember that you got the constants by integrating with respect to y. But because

\frac{\partial}{\partial y}f(x) = 0

, this integrating step gives you "constants" which can depend arbitrarily on x.

First question: Yes, same reason - your "constant" will be an arbitrary function of y.