- #1

- 26

- 3

**1. The problem statement, all variables, and given/known data**

Given is a second order partial differential equation $$u_{xx} + 2u_{xy} + u_{yy}=0$$ which should be solved with change of variables, namely ##t = x## and ##z = x-y##.

## Homework Equations

Chain rule $$\frac{dz}{dx} = \frac{dz}{dy} \cdot \frac{dy}{dx}$$

## The Attempt at a Solution

I was able to make the first change of variables $$\frac{\partial u}{\partial x} = \frac{\partial u}{\partial t} \cdot \frac{\partial t}{\partial x} + \frac{\partial u}{\partial z} \cdot \frac{\partial z}{\partial x} = u_t + u_z$$ but I'm stuck at making the second change of variables (for second derivative). If I attempt repeating the same process I end up with $$\frac{\partial^2 u}{\partial x^2} = \frac{\partial}{\partial x}(\frac{\partial u}{\partial t} \cdot \frac{\partial t}{\partial x} + \frac{\partial u}{\partial z} \cdot \frac{\partial z}{\partial x}) = \frac{\partial}{\partial x} \cdot \frac{\partial u}{\partial t} + \frac{\partial}{\partial x} \cdot \frac{\partial u}{\partial z}$$I don't think that I can write something like ##u_{tx}##?

I think this task is easy to solve, I'm apparently just missing something, Can somebody please help me? I don't expect anyone to solve homework tasks instead of me, I just need some guidance.

PS: Hope you're having a wonderful Tuesday!