S.O.S. Mathematics CyberBoard

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.

In this case note that x is constant as far as varying t is concerned, so it is just a collection of first-order linear ODE where you introduce a parameter x.

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