very simple pde advice
 

Greetings, I am not a maths graduate but have recently started a PhD in financial mathematics (I have an economics background). I am working through some notes provided by my superviser and have, I think, a working grasp of partial derivative equations.

I am a little confused on something and was wondering if someone would please advise me how to fix it.

I am trying to solve an equation

Pdu/dt = -Pda/dx -Pdb/dx

{where I am using notation Pda/dx to mean partial derivative of a with respect to x. The functions u,a and b are functions of x and t. The functions u, a and b are all equal to zero when x=0}

This equation comes from a simple fluid problem (which is why i thought this would be a good place to get help) and I require an analytical solution prior to doing some simple numerical solvers.

Am I correct in thinking that I can integrate both sides of the equation (even though one side is with respect to x and one side respect to t) and introduce an arbitrary function f(x,t) which, according to the boundary condition u=a=b=0 at x=0 would be equal to zero?

this would provide a solution u+a+b=0

if someone could advice me I'd be very happy!