# Partial Differential Equations

Can anyone help with these problems? I have no idea where to start. What is the general approach?

Determine the solution of ∂ρ/∂t = (sin x)ρ which satisfies ρ(x,0) = cos x.
Determine the solution of ∂ρ/∂t = ρ which satisfies ρ(x,t) = 1 + sin x along x =-2t.

Relevant equations: ∂ρ/∂t + ∂/∂x(q(ρ)) = 0 or ∂ρ/∂t + ∂/∂x(ρu(ρ)) = 0 and q = ρu.

Last edited:

Fredrik
Staff Emeritus
Gold Member
I suck at differential equations, but I think it's pretty clear that the approach should be to solve these equations as if they were ordinary differential equations (since they contain no derivatives with respect to x). The "constants" that appear in the solutions can of course depend on x, so instead of writing e.g. A, you write A(x), where A is a function. Then you use the "which satisfies..." statements to find A.

So how would I solve them? Which method should I use?

Fredrik
Staff Emeritus