Lecture Notes in Differential Equations - Bruce E. Shapiro

93
because the integral is zero (the top and bottom limits are identical).
Changing the label φ to y in equation (11.11) returns the second line of
(11.1), thus completing the proof of part (b).
The Method of Successive Approximations, which is also called Picard
Iteration, attempts to find a solution to the initial value problem
(11.1) by solving the integral equation (11.2). This will work because both
equations have the same solution. The problem is that solving the integral
equation is no easier than solving the differential equation.
The idea is this: generate the sequence of functions φ 0 , φ 1 , φ 2 , . . . , defined
by
φ 0 (t) = y 0 (11.12)
φ 1 (t) = y 0 +
φ 2 (t) = y 0 +
φ n+1 (t) = y 0 +
.
.
∫ t
t 0
f(s, φ 0 (s))ds (11.13)
∫ t
t 0
f(s, φ 1 (s))ds (11.14)
∫ t
t 0
f(s, φ n (s))ds (11.15)
From the pattern of the sequence of functions, we try to determine
φ(t) = lim
n→∞ φ n(t) (11.16)
If this limit exists, then it converges to the solution of the initial value
problem.
Theorem 11.2. Suppose that f(t, y) and ∂f(t, y)/∂y are continuous in
some box
then there is some interval
t 0 − a ≤ t ≤ t 0 + a (11.17)
y 0 − b ≤ y ≤ y 0 + b (11.18)
t 0 − a ≤ t 0 − h ≤ t ≤ t 0 + h ≤ t 0 + a (11.19)