HOMEWORK 2
Due Sept 13
1. It is possible to prove existence directly using nothing more than calculus.
¯ β) and bounded
Suppose that f (t, x) is defined in [t0 − α, t0 + α] × B(a,
with bound, M . Let b = min(α, β/M ). f has a Lipschitz constant L. Let
Z
t
yn+1 = a +
f (s, yn (s)) ds
t0
with y0 = a. Prove by induction that
|yn+1 (t) − yn (t)| ≤
Conclude that
a+
∞
X
M Ln (t − t0 )n+1
(n + 1)!
[yn+1 (t) − yn (t)] = y(t)
n=0
is uniformly convergent on [t0 − b, t0 + b], that is
y(t) = lim yn (t)
n→∞
exists uniformly. Show that y(t) is a solution to
Z
t
y(t) = a +
f (s, y(s)) ds.
t0
2. Solve the integral equation:
Z
x(t) = 1 +
t
sx(s) ds.
0
Compute the first few terms for the Picard iteration and then use this to
write the general nth iterate. Show that the terms are same as the Taylor
series expansion for your solution to the integral equation.
3. Find the maximal interval of existence to the ODE:
ẋ = x2 ,
1
x(0) = a.