partial differential equation

• May 13th 2011, 01:40 PM
mlazos
partial differential equation
Hi, I have the differential equation
$\frac{ \partial z }{ \partial x } + \frac{ \partial z }{ \partial y } = z^2$. Satisfied by $z(x,y)$, where $z = f(x)$ on y=0 and x is between minus to plus infnity, show that the solution is
$z = \frac{f(s)}{1-f(s) t}$, where x=t+s and y=t.

I know I have to use the chain rule but I am not sure how to start. Do you have any hints you could give me to start working the solution? Thanks a lot.
• May 13th 2011, 02:04 PM
topsquark
Quote:

Originally Posted by mlazos
Hi, I have the differential equation
$\frac{ \partial z }{ \partial x } + \frac{ \partial z }{ \partial y } = z^2$. Satisfied by $z(x,y)$, where $z = f(x)$ on y=0 and x is between minus to plus infnity, show that the solution is
$z = \frac{f(s)}{1-f(s) t}$, where x=t+s and y=t.

I know I have to use the chain rule but I am not sure how to start. Do you have any hints you could give me to start working the solution? Thanks a lot.

Probably the simplest approach would be to solve x = t + s, y = t for t and s in terms of x and y. Then plug that into your solution and you've got z(x, y). Taking the partials should be easy now.

-Dan
• May 13th 2011, 02:14 PM
mlazos
So we just need to verify that it is a solution or we maybe there is a way to find the solution using a method or something? If I show that the solution satisfies the partial differential equation is it sufficient? Thanks again.
• May 13th 2011, 03:03 PM
topsquark
Quote:

Originally Posted by mlazos
So we just need to verify that it is a solution or we maybe there is a way to find the solution using a method or something? If I show that the solution satisfies the partial differential equation is it sufficient? Thanks again.

My bad. I thought all you had to do was verify. Your equation is a first order quasilinear equation. One way to solve it is to use the method of characteristics which from the form of the solution was used here.

-Dan