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# Elementary Differential Equations and Boundary Value Problems - Boyce W.E.

Boyce W.E. Elementary Differential Equations and Boundary Value Problems - John Wiley & Sons, 2001. - 1310 p.
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where g(t) is large during a short interval t0 — ò < t < t0 + ò and is otherwise zero. The integral I(ò), defined by
ã (0+ò
I(ò) = g(t) dt, (2)
' l0 — x
or, since g( t) = 0 outside of the interval (t0 — ò, t0 + ò),
/ÒÎ
g(t) dt, (3)
•TO
is a measure of the strength of the forcing function. In a mechanical system, where g(t) is a force, I(ò) is the total impulse of the force g(t) over the time interval (t0 — ò, t0 + ò). Similarly, if y is the current in an electric circuit and g(t) is the time derivative of the voltage, then I (ò) represents the total voltage impressed on the circuit during the interval (t0 — ò, t0 + ò).
6.5 Impulse Functions
325
In particular, let us suppose that t0 is zero, and that g(t) is given by
g(t) = d (t) = ( )/lx' -t < t <T' (4)
®w T ’ j 0, t <-t or t > t, w
where t is a small positive constant (see Figure 6.5.1). According to Eq. (2) or (3) it follows immediately that in this case I (t) = 1 independent of the value of t , as long as t = 0. Now let us idealize the forcing function dT by prescribing it to act over shorter and shorter time intervals; that is, we require that t — 0, as indicated in Figure 6.5.2. As a result of this limiting operation we obtain
lim dT(t) = 0, t = 0. (5)
t —0 T
Further, since I(t) = 1 for each t = 0, it follows that
lim I(t) = 1. (6)
T —^ 0
Equations (5) and (6) can be used to define an idealized unit impulse function S, which imparts an impulse of magnitude one at t = 0, but is zero for all values of t other than zero. That is, the “function” S is defined to have the properties
S(t) = 0, t = 0; (7)
/TO
S(t) dt = 1. (8)
-TO
1
² 2ò I
-ò ò t
FIGURE 6.5.1 Graph of y = dT (t).
FIGURE 6.5.2 Graphs of y = dT (t) as t — 0.
326
Chapter 6. The Laplace Transform
There is no ordinary function of the kind studied in elementary calculus that satisfies
both of Eqs. (7) and (8). The “function” S, defined by those equations, is an example of
what are known as generalized functions and is usually called the Dirac2 delta function. Since S(t) corresponds to a unit impulse at t = 0, a unit impulse at an arbitrary point t = t0 is given by S(t - t0). From Eqs. (7) and (8) it follows that
S(t - t0) = 0> t = 0 (9)
/TO
S(t - t0) dt = 1. (10)
-TO
The delta function does not satisfy the conditions of Theorem 6.1.2, but its Laplace transform can nevertheless be formally defined. Since S(t) is defined as the limit of dT(t) as ò — 0, it is natural to define the Laplace transform of S as a similar limit of the transform of dT. In particular, we will assume that t0 > 0, and define L{S(t - t0)} by the equation
L{S(t - t0)}= lim C{dx(t - Q}. (11)
ò —— 0
To evaluate the limit in Eq. (11) we first observe that if ò < ^ which must eventually be the case as ò — 0, then t0 - ò > 0. Since dx(t - t0) is nonzero only in the interval from t0 - ò to t0 + ò, we have
Ï TO
L{dr(t - t0)} = e~stdr(t - t0) dt
0
t
*0+ò
e~std(t - L) dt.
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