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Introduction where c is an arbitrary constant of integration. The object starts to fall when t = 0, so we know that x = 0 when t = 0. From Eq. (28) it follows that c = −245, so the distance the object has fallen at time t is given by x = 49t + 245e−t/5 − 245. (29) Let T be the time at which the object hits the ground; then x = 300 when t = T . By substituting these values in Eq. (29) we obtain the equation 49T + 245e−T /5 − 545 = 0. (30) The value of T satisfying Eq. 51 sec. At this time, the corresponding velocity vT is found from Eq.

12 Chapter 1. Introduction a particular value of c, and is the graph of the solution corresponding to that value of c. Satisfying an initial condition amounts to identifying the integral curve that passes through the given initial point. To relate the solution (18) to Eq. (2), which models the field mouse population, we need only replace a by the growth rate r and b by the predation rate k. Then the solution (18) becomes p = (k/r ) + [ p0 − (k/r )]er t , (19) where p0 is the initial population of field mice.

Equation (5) expresses a relation between the independent variable t and the values of the function u and its first n derivatives u , u , . . , u (n) . It is convenient and customary in differential equations to write y for u(t), with y , y , . . , y (n) standing for u (t), u (t), . . , u (n) (t). Thus Eq. (5) is written as F(t, y, y , . . , y (n) ) = 0. (6) y + 2et y + yy = t 4 (7) For example, is a third order differential equation for y = u(t). Occasionally, other letters will be used instead of t and y for the independent and dependent variables; the meaning should be clear from the context.