I know how to solve linear first order partial differential equations with two independent variables using the charactereristics method.
My question is: How to solve firts order linear PDE if it contains three independent variables: $$a(x,y,z)u_x+b(x,y,z)u_x+c(x,y,z)u_z=0$$
How to solve (with the characteristic method) first order linear PDE if it contains three independent variables: $$a(x,y,z)u_x+b(x,y,z)u_x+c(x,y,z)u_z=0$$ Obviously, first learn a course about the characteristic method.
Then solve the characteristic ODE equations : $$\frac{dx}{a(x,y,z)}=\frac{dy}{b(x,y,z)}=\frac{dz}{c(x,y,z)}$$ Solving $\quad \frac{dy}{dx}=\frac{b(x,y,z)}{a(x,y,z)}\quad$ leads to an implicite solution : $\quad f(x,y,z)=C_1$
Solving $\quad \frac{dz}{dx}=\frac{c(x,y,z)}{a(x,y,z)}\quad$ leads to an implicite solution : $\quad g(x,y,z)=C_2$
The general solution of the PDE is: $$u(x,y,z)=F\left(f(x,y,z), g(x,y,z)\right)$$ where $F$ is any differentiable function of two variables.