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.