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$\displaystyle \frac{\partial u}{\partial x}+\frac{\partial u}{\partial y}=1$

Scratch work:

I know how to solve the HOMOgeneous PDE: $\displaystyle \frac{\partial u}{\partial x}+\frac{\partial u}{\partial y}=0$

The LHS is just the directional derivative of $\displaystyle u$ in the direction of $\displaystyle [1, 1]$.

So if a curve in 3D has $\displaystyle [1,1]$ as a tangent vector, the curve's equation is just $\displaystyle \frac{dy}{dx} = 1$

so $\displaystyle y = x + C_1$ so $\displaystyle x - y = -C_1 = C$. So $\displaystyle u(x,y) = f(x - y)$, where $\displaystyle f$ is differentiable.

But how do I solve the NONhomogeneous question from this?

Thanks everybody.