文档介绍：Chapter 1
Existence and Uniqueness
Introduction
An ordinary differential equation (ODE) is given by a relation of the form
F (t, x, x0, x00, · · · , x(m)) = 0 , ()
where t ∈ R, x, x0, · · · , x(m) ∈ Rn. The function F is defined in an open set of
R × Rn × · · · × Rn. We say that the ODE is of order m if the maximal order of the
derivative occurring in () is m. A function x : I → Rn, where I is an interval in R
is a solution of () if x(t) is of class Cm (. m-times continuously differentiable) and
if
F (t, x(t), x0(t), x00(t), · · · , x(m−1)(t)) = 0 for all t ∈ I . ()
Example Clairaut equation (1734) Let us consider the first order equation
x − tx0 + f(x0) = 0 , ()
where f is some given function. It is given in implicit form by a nonlinear equation in
x0. It is easy to verify that the lines x(t) = Ct − f(C) are solutions of () for any
2
C. For example consider f(z)z + z. One sees easily that given a point (t0, x0) there
exists 0 or 2 solutions passing by the point (t0, x0).
As we see from this example, it is in general very difficult to obtain results on the
uniqueness or existence of solutions for general equations of the form (). We will
restrict ourselves to situations where () can be solved as a function of x(m),
x(m) = g(t, x, x0, · · · , x(m−1)) , ()
Such an equation is called explicit. If we introduce the new variables
0 00 (m−1)
x1 = x , x2 = x , x3 = x · · · , xm = x , ()
CHAPTER 1. EXISTENCE AND UNIQUENESS 2
then we can rewrite () as the system
0
x1 = x2 ,
0
x2 = x3 ,
.
. ()
0
xm−1 = xm ,
0
xm = g(t, x1, x2, · · · , xm) .
nm
This is an equation of order 1 for the supervector x = (x1, · · · , xm) ∈ R (each xi is
in Rn) and it has the form x0 = f(t, x). Therefore, in general, it is sufficient to consider
the case of first order equations (m=1).
Example Predator-Prey equation Let us consider the equation
x0 = x(α−βy) , y0 = y(γx −δ) , ()
where α, β, γ