I'm doing revision for an exam in 2 days and I don't quite understand one of the final steps in the problem (worked example)

subject to

Solution

Parametrization of IC's ->

I don't quite understand why this is done. I can see how its used to solve this question....is it just a standard method to solve PDE's?

**This is standard to get an ODE.**
Then

Apply the initial data to get

I'm not sure why this is....

**You took the logarithm of both sides (you should be worried in a PDE class if this is not clear).**
and

PDE becomes

and I don't understand any of the remaining steps:

(I'm not sure where

comes from and the RHS is just integrating with respect to t, right?

**You took the integral of both sides (first method you learn to solve a DE in Calculus 1). The left side is clearly with respect to u and the right side is with respect to t.** Im not sure what

is or where the

term comes from

**You exponentiated both sides and the D(s) comes from the fact that your constant C is a function of s (which might just be a real number).** . I don't understand this because I don't know what

is.

**That's why you plugged in t=0, it gives you that D(s)=F(s), where F(s) is given in the problem.** I actually think I understand this line.

Can someone please explain this to me? I'm worried.