Help on solving first-order NONhomogeneous PDE

Hello everybody. I'm trying to solve this first-order NONhomogenous PDE. But because my awfully bad textbook only gave examples on how to solve homogeneous PDEs, I'm not sure what to do. Here's my scratch work.

I also know the theorem that if two solutions solve linear NONhomogeneous PDE, then their difference solve linear homogeneous PDE. But I don't think that's going to help here.

**Problem: Solve **

**Scratch work:**

I know how to solve the HOMOgeneous PDE:

The LHS is just the directional derivative of in the direction of .

So if a curve in 3D has as a tangent vector, the curve's equation is just

so so . So , where is differentiable.

But how do I solve the NONhomogeneous question from this?

Thanks everybody.

Re: Help on solving first-order NONhomogeneous PDE

One solution is u = x. If you let you'll end up with .

BTW - do you know the method of characteristics?

Re: Help on solving first-order NONhomogeneous PDE

Quote:

Originally Posted by

**Danny** One solution is u = x. If you let

uou'll end up with

.

BTW - do you know the method of characteristics?

Hello and thanks Danny. So could've been just guessed?

But if , wouldn't you get ? Since because there is no *y* in here.

And I don't think so since I'm taking a first course in PDEs.

Re: Help on solving first-order NONhomogeneous PDE

Well, you assme that . The method of characteristics will probably come up. It's a way to solve

.

Re: Help on solving first-order NONhomogeneous PDE

Hello and thanks Danny.

Quote:

Originally Posted by

**Danny** Well, you assme that

.

But when you say here, do you mean and not just any which I found in my original post for the homogeneous PDE?

And did I get right?

Re: Help on solving first-order NONhomogeneous PDE

Well, you would end up with that in the end. Also, if you let then

gives gives .