Determine p(z) by solving the differential equation dp/dz = -1/λ*p where λ is a constant, and find the particular solution that satisfies the initial condition p(0) = P , where P is a constant.
I've assumed that using seperation of variables method is suitable as the RHS is a multiple of two functions, is this correct and if so have i approached it in the right way?
1/p dp = λ^1/p dz
Integrate both sides
ln|p|= 1/λ*ln|p|+C
Assuming that $p$ is in the numerator on the RHS: \begin{align} \frac{dp}{dz} &= -\frac1\lambda p \\ \frac{dp}p&=-\frac{dz}\lambda \\ \ln p &=-\frac z\lambda + C \\ p(z) &= p(0)e^{-\frac z\lambda} \end{align}
Assuming that $p$ is in the denominator on the RHS: \begin{align} \frac{dp}{dz} &= -\frac1{\lambda p} \\ pdp&=-\frac{dz}\lambda \\ \frac12 p^2 &=-\frac z\lambda + C \\ p(z) &= \sqrt{p(0)^2-\frac{2z}\lambda} \end{align}