ABSTRACT
In this chapter, we will learn about nonlinear differential equations. In engineering often our first mathematical model is linear but a more accurate model is nonlinear. In general, we have an initial value problem (IVP), that is, a system of differential
equations and initial condition { x˙ = f(x, t) x(t0) = x0
} , (18.1)
and we will assume that we have existence and uniqueness of a solution to the IVP, meaning that given an initial time t0 and an initial value x(t0), (18.1) has exactly one solution x(t) for t in some interval (t0 − δ, t0 + δ) with δ > 0. This is the generalization to systems of the results in Section 3.2. In fact, let us assume that f(x, t) is continuously differentiable in x, which is enough to guarantee the existence and uniqueness property. We will say more about this in Section 18.7. Because of the uniqueness, we can notate that the solution of (18.1) is x(t)= x(t; t0, x0),
that is, we acknowledge that the solution of IVP (18.1) depends on the initial condition. In the special case of initial time t0 = 0, we notate x(t; t0, x0)= x(t; x0); the omission of t0
will imply that the initial time is t0 = 0. Note that x(0; x0)= x0. We saw in Chapters 3 and 4 that if x(t)= y(t) is one dimensional (1D), that is, x is in R1,
we can show a solution of (18.1) as a graph in the (t, y)-plane. For example, in Figure 18.1 we show a graph of the solution of y˙+ 10y= − 2 sin 2t, y(0)= 20/13, which illustrated the concept of a steady-state oscillation. Similarly, x(t) in R2 can be graphed in (t, x1(t), x2(t))-space. But suppose the system
is autonomous, meaning that the right-hand side of the system is f(x), which does not depend on t, that is, we are studying the IVP
{ x˙ = f(x) x(t0) = x0
} . (18.2)
Then, in a sense, all initial times t0 are the same.
