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

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In each of Problems 8 through 11 find the inverse Laplace transform of the given function by using the convolution theorem.

1s

8. F(s) = -— 9. F(s) =

s4(s2 + 1) ' (s + 1)(s2 + 4)

10. F(s) = , . * 2 11. F(s) =

(s + 1)2(s2 + 4) s2 + 1

3This terminology arises from the fact that H(s) is the ratio of the transforms of the output and the input of the problem (27).

336

Chapter 6. The Laplace Transform

In each of Problems 12 through 19 express the solution of the given initial value problem in terms of a convolution integral.

12. / + â2y = g(t); y(0) = 0, /(0) = 1

13. Ó' + 2 Ó + 2 y = sin at; y(0) = 0, Ó (0) = 0

14. 4Ó' + 4/+ 17y = g(t); y(0) = 0, /(0) = 0

15. Ó + Ó + 5y = 1 — ujt); y(0) = 1, /(0) = -1

16. Ó + 4y + 4y = g(t); y(0) = 2, /(0) = -3

17. Ó + 3y + 2y = cos at; y(0) = 1, /(0) = 0

18. yv — y = g(t); y(0) = 0, y(0) = 0, /(0) = 0, /'(0) = 0

19. yv + 5Ó + 4y = g(t); y(0) = 1, /(0) = 0, /(0) = 0, /'(0) = 0

20. Consider the equation

in which f and k are known functions, and ô is to be determined. Since the unknown function ô appears under an integral sign, the given equation is called an integral equation; in particular, it belongs to a class of integral equations known as Volterra integral equations. Take the Laplace transform of the given integral equation and obtain an expression for Ñ{ô(´)} in terms of the transforms L{ f (t)} and L{k(t)} of the given functions f and k. The inverse transform of Ñ{ô(¥)} is the solution of the original integral equation.

21. Consider the Volterra integral equation (see Problem 20)

(b) Show that the given integral equation is equivalent to the initial value problem

(c) Solve the given integral equation by using the Laplace transform.

(d) Solve the initial value problem of part (b) and verify that the solution is the same as that obtained in part (c).

22. The Tautochrone. A problem of interest in the history of mathematics is that of finding the tautochrone—the curve down which a particle will slide freely under gravity alone, reaching the bottom in the same time regardless of its starting point on the curve. This problem arose in the construction of a clock pendulum whose period is independent of the amplitude of its motion. The tautochrone was found by Christian Huygens (16291695) in 1673 by geometrical methods, and later by Leibniz and Jakob Bernoulli using analytical arguments. Bernoulli’s solution (in 1690) was one of the first occasions on which a differential equation was explicitly solved.

The geometrical configuration is shown in Figure 6.6.2. The starting point P(a, b) is joined to the terminal point (0, 0) by the arc C. Arc length s is measured from the origin, and f (y) denotes the rate of change of s with respect to y:

Then it follows from the principle of conservation of energy that the time T(b) required for a particle to slide from P to the origin is

(a) Show that if u is a function such that u"(t) = ô(¥), then

u"(t) + u(t) — tu'(0) — u(0) = sin2t.

u" (t) + u(t) = sin2t; u(0) = 0, u'(0) = 0.

(i)

(ii)

6.6 The Convolution Integral

REFERENCES

(a) Assume that T(b) = T0, a constant, for each b. By taking the Laplace transform of Eq. (ii) in this case and using the convolution theorem, show that

F(s) =

then show that

f (y) = J — -0=

2_g —_

Hint: See Problem 27 of Section 6.1.

(b) Combining Eqs. (i) and (iv), show that

dx

dy"

2à - y

y ’

where à = gT^/n2.

(c) Use the substitution y = 2à sin2 (â/2) to solve Eq. (v), and show that x = à(â + sin â), y = à(1 - cos â).

(iii)

(iv)

(v)

(vi)

Equations (vi) can be identified as parametric equations of a cycloid. Thus the tautochrone is an arc of a cycloid.

The books listed below contain additional information on the Laplace transform and its applications: Churchill, R. V., Operational Mathematics (3rd ed.) (New York: McGraw-Hill, 1971).

Doetsch, G., Nader, W. (tr.), Introduction to the Theory and Application of the Laplace Transform (New York: Springer-Verlag, 1974).

Kaplan, W., Operational Methods for Linear Systems (Reading, MA: Addison-Wesley, 1962). Kuhfittig, P. K. F., Introduction to the Laplace Transform (New York: Plenum, 1978).

Miles, J. W., Integral Transforms in Applied Mathematics (London: Cambridge University Press, 1971).

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