I have done most of the questions, I am just struggling on a few parts. The question is:

a) Use the method of Separation of Variables to find the General Solution.

b) Analyse the right hand side and determine for what pairs

the initial value problems (IVP) would be guaranteed to have a solution by Peano's theorem, and a unique solution by Picard's theorem.

c) Find solutions satisfying the specified initial conditions, and determine the intervals where they exists and where they are unique.

1)

. IC = y(1) = 0.

Answers:

a) I got:

, which was correct. But for some reason, they said this was the "typical solution" and I also needed a "special solution" where

. My first question is:

*What does the special solution do?*
b) I know Picard's and Peano's theorem, so I checked where the RHS was continuous and everything and I got that the RHS of my IVP is continuous for all (x,y), and the partial derivative w.r.t 'y' is continuous everywhere except y = 0, so the two intervals I need are:

The solutions to the IVP exist for any pair

(but for some reason the answers said it is contained in R multiplied by R. Why?), and the unique solution will exists for all values of

, but not when

and so the interval would be

, again, they however multiplied both intervals by some 'R'.

I understand this, I just don't get why they multiplied everything by that 'R', it looks like an R for the set of Real numbers, but I don't see why you need to multiply it by this.

c) This is the main one I am having trouble with. This is what it says in the answers:

The initial condition is satisfied for the special solution y = 0 (

* Again, what does this mean? What would happen if it didn't satisfy the special solution? *). For the typical solution, we have

, hence C = 0 and

. This general form is a "compound" solution (

*Whatever that means*)of the form

y = {

or

depending on one arbitary parameter

. Each of these solutions is defined for all

, and they are NOT UNIQUE IN ANY INTERVAL, since B is not uniquely defined.

Apart from the special solution bit, I get the bit up to C = 0 and your "new" equation, but I don't understand it after that. How do you know the solution is in that other form? How do I solve an equation like that?

Thank you