[note: Ux=∂U/∂x, Uy=∂U/∂y] Example: Solve the partial differential equation 2Ux + 3Uy + U = 0 by using the change of variables V(x,y)=ln[U(x,y)] Solution: Vx = Ux/U Vy = Uy/U 2Ux + 3Uy + U = 0 Dividing both sides by U, we have 2Ux/U + 3Uy/U + 1 = 0 => 2Vx + 3Vy +1 = 0 => 2Vx + 3Vy = -1 The corresponding homogeneous equation has the general solution V = f(3x-2y) where f is arbitrary function. 2Vx + 3Vy = -1 Set V(x,y)=f(x) => 2f ' + 0 = -1 => f ' = -1/2 => f= -x/2 + C => f= -x/2 (take C=0) Therefore, a particular solution to 2Vx + 3Vy = -1 is V = -x/2 So the general solution to 2Vx + 3Vy = -1 is V = -x/2 + f(3x-2y) => the general solution to the original PDE is U = exp(-x/2) exp[f(3x-2y)] = exp(-x/2) g(3x-2y) =============================== Now, I don't understand the parts in red: 1) In the solution, they divided both sides by U. Why is this always allowed? How do we know that U is not 0? 2) For the part of finding a particular solution to 2Vx + 3Vy = -1, they first set V(x,y)=f(x). What is the logic behind this step? Why would this lead us to a particular solution? I just don't get the idea. 3) At the end, they claimed that the general solution to the original PDE is U = exp(-x/2) exp[f(3x-2y)] i.e. U= exp(-x/2) g(3x-2y) (final answer) I don't understand why exp[f(3x-2y)] can be replaced by g(3x-2y). Why do we have to do that? Is g here an arbitrary function? Could someone please kindly explain? Any help is greatly appreciated!