Solving a 3D PDE with given initial conditions and characteristics"

[Easty]

b]1. Homework Statement [/b]
Find the characteristics, and then the solution, of the partial differential equation

x\frac{\partial u}{\partial x}+xy\frac{\partial u}{\partial y}+z\frac{\partial u}{\partial z}=0


given that u(1, y, z)=yz

Homework Equations

The Attempt at a Solution

Found this question on an old exam and am not quite sure what to do.
Initialy i tried to take the z derivative to the other side and still take dy/dy=y to be the characteristics, leaving me to solve

x\frac{\partial u}{\partial x}+z\frac{\partial u}{\partial z}=0.

but i think that's wrong.
Could someone please point me in the right direction.
thanks
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[HallsofIvy]

Are you clear on what "characteristics" are for a partial differential equation?

The characteristics for this equation are given by
dx/dt= x, dy/dt= xy, and dz/dt= z. From the first and third, x= Ae^t and z= Ce^t. The second equation is dy/dt= Ae^ty which separates as dy/y= Ae^tdt and integrates as ln(y)= Ae^t+ B' or y= Be^Aet= Be^x. Also, z= Ce^t and, from x= Ae^t, e^t= x/A so z= Ce^x/A. That is, taking x= t as parameter, the charateristics are given by x= t, y= Be^t, and z= Ce^t/A.

 

[Easty]

Ok thanks. So would the solution of the PDE still be just an arbitary function of the constant combination of variables? how do i combine all the characteristics to form a single solution?