## CdU U1 U263

Since this is a second order equation, we need two conditions to specify its solution. With reference to Figure 2.21b, recall that we are thinking about an infinite spatial domain but a finite time domain. Also, with reference to that figure, at any time, as x !—1, U approaches 1 and as x !i, U approaches 0. We thus have the conditions that U(—i) = 1 and U(i) = 0 to go along with Eq. (2.63).

We are not going to try to solve Eq. (2.63), but we will succeed in analyzing it. The first step in this analysis is to convert it to a system of ordinary differential equations by introducing W as the derivative of U:

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