Lecture Notes in Differential Equations - Bruce E. Shapiro

Lesson 11
Method of Successive
Approximations
The following theorem tells us that any initial value problem has an equivalent
integral equation form.
Theorem 11.1. The initial value problem
}
y ′ (t, y) = f(t, y)
y(t 0 ) = y 0
(11.1)
has a solution if and only if the integral equation
has the same solution.
φ(t) = y 0 +
∫ t
0
f(s, φ(s))ds (11.2)
Proof. This is an “if and only if” theorem, so to prove it requires showing
two things: (a) that (11.2) =⇒ (11.1); and (b) that (11.1) =⇒ (11.2).
To prove (a), we start by assuming that (11.1) is true; we then need to
show that (11.2) follows as a consequence.
If (11.1) is true then it has a solution y = φ(t) that satisfies
⎫
dφ
= f(t, φ(t)) ⎬
dt
⎭
φ(t 0 ) = y 0
(11.3)
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