Differential Equations and Linear Algebra, 4th Edition

A.3 Local Existence

In the case of a nonlinear initial value problem

$$\frac{d\mathbf{\text{x}}}{dt}=\mathbf{\text{f}}(\mathbf{\text{x}},t),\mathbf{\text{x}}(a)=\mathbf{\text{b}},$$ (35)

the hypothesis in Theorem 1 that f satisfies a Lipschitz condition on a slab (x, t) (t in I, all x) is unrealistic and rarely satisfied. This is illustrated by the following simple example.

Example 1

Consider the initial value problem

$$\frac{dy}{dx}=x^2,x(0)=b>0.$$ (36)

As we saw in Example 6, the equation $x\text{′}=x^2$ does not satisfy a “strip Lipschitz condition.” When we solve (36) by separation of variables, we get

$$x(t)=\frac{b}{1-bt}.$$ (37)

Because the denominator vanishes for $t=1/b,$ Eq. (37) provides a solution of the initial value problem in (36) only for $t<1/b,$ despite the fact that the differential equation $x\text{′}=x^2$ “looks nice” on the entire real line—certainly the function ...