Find an interval centered about x=0 for which the given IVP has a unique solution:
(x-2)y''+3y=x, y(0)=0, y'(0)=1
what exactly is the question asking?? Do I need to solve the IVP to find the boundaries?
Well, you could solve it then prove that this solution is unique (not too difficult, I think, since it's a linear equation). Or you could use the existence and uniqueness theorem for linear equations that says that if where are continous functions of on then there is a unique solution to this equation a,b) \rightarrow \mathbb{R}" alt="ya,b) \rightarrow \mathbb{R}" /> with the specified initial conditions on a point.
If we suppose that the solution we are searching is analitic in we can write...
(1)
From (1) and the 'initial conditions' we derive that is...
(2)
... and the DE can be written as...
(3)
If we impose the identity (3) for all we obtain first...
(4)
... and for ...
(5)
The conlusion is that the solution can be written as...
(6)
... and because the sequence in (6) is bounded the series in (6) will converge if ...
Kind regards