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## Homework Statement

Find the general solution f = f(x,y) of class C

^{2}to the partial differential equation

[tex]\frac{\partial^2 f}{\partial x^2}+4\frac{\partial^2 f}{\partial x \partial y}+\frac{\partial f}{\partial x}=0[/tex]

by introducing the new variables u = 4x - y, v = y.

## Homework Equations

With "by introducing the new variables" I assume they mean something like this

[tex]f=f(x,y)=f(u(x,y), y)=f(4x-y, y)[/tex]

[tex]\frac{\partial u}{\partial x}=u_1(x,y)=u_1=4[/tex]

[tex]\frac{\partial u}{\partial y}=u_2(x,y)=u_2=-1[/tex]

Which means the second derivatives of u = 0.

## The Attempt at a Solution

I'm not sure if this is the right way to approach the problem, but I'll show what I got so far.

I calculated the partial derivatives according to the first equation and ended up with

[tex]f_{11}u_{1}^{2}+4(f_{11}u_{1}u_{2}+f_{21}u_{1})+f_{1}u_{1}=0[/tex]

u

_{1}= 4, u

_{2}= -1 gives

[tex]4f_{21}+f_{1}=4\frac{\partial^2 f}{\partial x \partial y}+\frac{\partial f}{\partial x}=0[/tex]

From here, I'm uncertain of how to proceed (if I even used the right method to being with).

**We haven't gone through PDE's in our calculus course yet (just going through the basics of multivariable calculus at the moment), so I'm only going to assume this can be solved without much/any experience with PDE's.**

__NOTE:__