# Finding Intervals of Solutions to ODE's

1. Feb 7, 2013

### BrainHurts

1. The problem statement, all variables and given/known data

Consider the IVP

$\frac{dy}{dt}$ = t2 + y2, y(0)=(0)

and let B be the rectangle [0,a] x [-b,b]

a) the solution to this problem exists for

0≤t≤min{a, $\frac{b}{a2+b2}$

b) that min{a,$\frac{1}{2}$a} is largest when a=$\frac{1}{\sqrt{2}}$
c) Deduce an interval 0≤t≤α on which the solution to this problem exists and is unique.

2. Relevant equations

3. The attempt at a solution

for a) f(t,y)= t2 + y2

the local Lipschitz condition $\frac{∂f}{∂y}$ = 2y is continuous for all (t,y)

so M=max(t,y)$\inB$|f(t,y)| = a2+b2

and from what we did i see the b/b2+a2

it's really the other two questions that I'm really confused on. Part b) and c). Any help would be great!