1550078481-Ordinary_Differential_Equations__Roberts_

272 Ordinary Differential Equations

y(x) = ~u(x - 2)e-(x-^2 ) sin 2(x - 2) + 4e-x cos 2x + 2e-x sin 2x

2

{

4e-x cos 2x + 2e-x sin 2x.

= ~e-(x-^2 ) sin 2(x - 2) + 4e-x cos 2x + 2e-x sin 2x, 2<x

A graph of this solution is displayed in Figure 5.9. The portion of the graph to
the right of x = 2 is due to the impulse force which was applied to the system
at the instant x = 2. The "solution" y(x) is continuous for x 2:: 0 but the first

derivative has a jump discontinuity at x = 2 and the second derivative has

an infinite discontinuity there. The dotted curve appearing in Figure 5.9 for

x > 2 shows the solution, if no impulse force were applied to the system at

the instant x = 2.

Figure 5.9 Graph of the Solution

y(x) =^5

2

u(x - 2)e-(x-^2 ) sin2(x - 2) + 4e-x cos 2x + 2e-x sin2x