# PDE of a 3d function

1. Jun 18, 2009

### Easty

b]1. The problem statement, all variables and given/known data[/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

2. Relevant equations

3. 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 thats wrong.
Could someone please point me in the right direction.
thanks
[
1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution

2. Jun 18, 2009

### 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= Aet and z= Cet. The second equation is dy/dt= Aety which separates as dy/y= Aetdt and integrates as ln(y)= Aet+ B' or y= BeAet= Bex. Also, z= Cet and, from x= Aet, et= x/A so z= Cex/A. That is, taking x= t as parameter, the charateristics are given by x= t, y= Bet, and z= Cet/A.

3. Jun 18, 2009

### 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?