ODE Lecture Notes
Section 2.8
Page 1 of 4
Section 2.8: The Existence and Uniqueness Theorem
Big Idea: The heart of the proof of the fundamental existence and uniqueness theorem for firstorder initial value problems involves the use of a converging succession of approximations to the
solution.
Big Skill: You should be able to use the equivalent integral equation for a first-order differential
equation to compute successively better approximations to the solution of the equation.
One simplification to proving the existence and uniqueness theorem is that any initial
value problem y  f  t , y  , y  t0   y0 can be transformed to an equivalent initial value problem
  f  tnew , ynew  , ynew  0  0 by a change of variables, which is equivalent to a translation of
ynew
the coordinate axes. Thus, it is sufficient to consider y  f  t , y  , y  0   0 when proving the
existence and uniqueness of the solution of any first-order differential equation initial value
problem.
Practice:
 
1. Transform the initial value problem y  e y sin  t  , y     ln  2  into an equivalent
2
problem with the initial condition at the origin.
The existence and uniqueness Theorem can now be stated as:
Theorem 2.8.1: Existence and Uniqueness of the Solution of First-Order Initial Value
Problems
f
If f and
are continuous in a rectangle R: t  a, y  b , then there is some interval t  h  a in
y
which there exists a unique solution y    t  of the initial value problem y  f  t , y  , y  0   0 .
ODE Lecture Notes
Section 2.8
Page 2 of 4
Note that a differential equation relates a function to its derivative. It is possible to rewrite the differential equation in a form that instead relates the function to its integral. Such an
equation is called an integral equation. To perform this transformation, we suppose temporarily
that y    t  satisfies the initial value problem. Thus:
y  f  t , y 
d
 t   f t, t 
dt
t
  t    f  s,   s   ds
0
The differential equation is transformed to an integral equation because it is easier to show that
there is a unique solution of the integral equation in some interval t  h  a .
The way the book goes about proving there is a unique solution to the integral equation is known
as the method of successive approximations, or Picard’s iteration method. In this method, we
choose an initial function 0  t  to approximate the solution, plug it into the integral equation,
evaluate the integral to get a new function 1  t  , and then use that new function to get a better
approximate function, etc.:
t
1  t    f  s, 0  s   ds
0
t
2  t    f  s, 1  s   ds
0
…
t
n 1  t    f  s, n  s   ds
0
Each member of the sequence of functions n  t   1  t  , 2  t  ,... satisfies the initial condition,
but it is unlikely that any of them satisfy the differential equation. It is our hope that the limit of
the sequence converges to that solution…
ODE Lecture Notes
Section 2.8
Page 3 of 4
Practice:
2. Solve the initial value problem y   t  y , y(0) = 0 using Picard’s method of successive
approximations.
ODE Lecture Notes
Section 2.8
Page 4 of 4
Notes on using Picard’s technique to prove the existence and uniqueness theorem:
1. Have to show that all members of the sequence n  t  exist in the rectangle R… can
assure this by restricting the t values of R according to maximum value of f… see pages
116-117.
2. Have to show that the sequence n  t  converges… see Problems 15 – 18.
3. Have to show that the sequence n  t  converges to (t), which is the solution of the
equation… too hard a proof for this level…see pages 117-118 for a discussion.
4. Have to show that (t) is unique… see Problem 19.