The method of successive approximations
uses the equivalent integral equation for (1) and an iterative method
for constructing approximations to the solution. This is a
traditional way to prove (1) and appears in most all differential
equations textbooks. It is attributed to the French
mathematician Charles
Emile Picard (1856-1941).

**Theorem 2 (Successive Approximations -
Picard Iteration).** The solution to the I.V.P in
(1) is found by constructing recursively a
sequence of
functions

, and

(2)

.

Then the solution
to (1) is given by the limit:

(3) .

**Proof.**

Begin by reformulating (1) as an
equivalent integral equation. Integration of both sides of
(1) yields

(4) .

Applying the Fundamental
Theorem of Calculus to the left side of (4)
yields and
we have , which
can be rearranged to obtain

(5) * *

Observe that occurs
on both the left and right hand sides of equation (5). We
can use this formula and input in
the integrand
on the right and then output the next iteration
for on
the left side. This is a type of fixed
point iteration, the most familiar form of which is
Newton's
method for root finding.

Start the iteration with the initial
function

,

then define the next function as
follows

.

Next
is used to construct as
follows

.

The process is repeated, and once
has been obtained, the next function is given recursively
by

(6) .

We must take the limit
as in
(6). Assume that the limit (3) exists,
then and
we write

If there is "no problem" when taking limits on the right side then we
might expect the following

This is the "intuitive proof" of equation (5). Q.E.D.

**For More
Proof.** More details for the existence and
uniqueness of can
be found in textbooks and the literature.

- Stephen W. Goode, Differential Equations and Linear Algebra, 3rd edition, Prentice Hall, Upper Saddle River, NJ, 2005, Appendix 4.
- R. Kent Nagle; Edward B. Saff; and Arthur David Snider, Fundamentals of Differential Equations and Boundary Value Problems, 4th edition, Addison-Wesley, Boston, MA, 2004, Chapter 13, Sections 1 and 2.
- C. Henry Edwards; and David E. Penney, Differential Equations and Boundary Value Problems: Computing and Modeling, 3rd edition, Prentice Hall, Upper Saddle River, NJ, 2005, Appendix A.1.
- William E. Boyce; and Richard C. DiPrima, Elementary Differential Equations and Boundary Value Problems, 7th Edition, John Wiley and Sons, New York, NY, 2002, Chapter 2, Section 8.
- Garrett Birkhoff; and Gian-Carlo Rota, Ordinary Differential Equations, 4rd Ed., John Wiley and Sons, New York, NY, 1989, Chapter 1, p. 23.
- Einar Hille, Lectures on Ordinary Differential Equations, Addison-Wesley Pub. Co., Reading, MA, 1969, pp. 32-41.
- Arthur Wouk, On the Cauchy-Picard Method, The American Mathematical Monthly, Vol. 70, No. 2. (Feb., 1963), pp. 158-162, Jstor.

(c) John H. Mathews 2005