Full solution: In each of 13 through 20, express the solution of the given initial value

. Establish the commutative, distributive, and associative properties of the convolution integral. (a) f g = g f (b) f (g1 + g2) = f g1 + f g2 (c) f (g h) = (f g) h

Find an example different from the one in the text showing that (f 1)(t) need not be equal to f(t).

Show, by means of the example f(t) = sin t, that f f is not necessarily nonnegative.

In each of Problems 4 through 7, find the Laplace transform of the given function. f(t) = t 0 (t )2 cos 2 d

In each of Problems 4 through 7, find the Laplace transform of the given function. f(t) = t 0 e(t) sin d

In each of Problems 4 through 7, find the Laplace transform of the given function. f(t) = t 0 (t )e d 7

In each of Problems 4 through 7, find the Laplace transform of the given function. f(t) = t 0 sin(t ) cos d

In each of Problems 8 through 11, find the inverse Laplace transform of the given function by using the convolution theorem. F(s) = 1 s4(s2 + 1)

In each of Problems 8 through 11, find the inverse Laplace transform of the given function by using the convolution theorem. F(s) = s (s + 1)(s2 + 4)

In each of Problems 8 through 11, find the inverse Laplace transform of the given function by using the convolution theorem. F(s) = 1 (s + 1)2(s2 + 4)

In each of Problems 8 through 11, find the inverse Laplace transform of the given function by using the convolution theorem. F(s) = G(s) s2 + 1

(a) If f(t) = t m and g(t) = t n, where m and n are positive integers, show that f g = t m+n+1 1 0 um(1 u) n du. (b) Use the convolution theorem to show that 1 0 um(1 u) n du = m! n! (m + n + 1)! . (c) Extend the result of part (b) to the case where m and n are positive numbers but not necessarily integers.

In each of Problems 13 through 20, express the solution of the given initial value problem in terms of a convolution integral. y + 2y = g(t); y(0) = 0, y (0) = 1

In each of Problems 13 through 20, express the solution of the given initial value problem in terms of a convolution integral. y + 2y + 2y = sin t; y(0) = 0, y (0) = 0

In each of Problems 13 through 20, express the solution of the given initial value problem in terms of a convolution integral. 4y + 4y + 17y = g(t); y(0) = 0, y (0) = 0

In each of Problems 13 through 20, express the solution of the given initial value problem in terms of a convolution integral. y + y + 5 4 y = 1 u(t); y(0) = 1, y (0) = 1

In each of Problems 13 through 20, express the solution of the given initial value problem in terms of a convolution integral. y + 4y + 4y = g(t); y(0) = 2, y (0) = 3

In each of Problems 13 through 20, express the solution of the given initial value problem in terms of a convolution integral. y + 3y + 2y = cos t; y(0) = 1, y (0) = 0

In each of Problems 13 through 20, express the solution of the given initial value problem in terms of a convolution integral. . y(4) y = g(t); y(0) = 0, y (0) = 0, y(0) = 0, y(0) = 0

In each of Problems 13 through 20, express the solution of the given initial value problem in terms of a convolution integral. y(4) + 5y + 4y = g(t); y(0) = 1, y (0) = 0, y(0) = 0, y(0) = 0

Consider the equation (t) + t 0 k(t )() d = f(t), in which f and k are known functions, and is to be determined. Since the unknown function appears under an integral sign, the given equation is called an integral equation; in particular,it belongs to a class of integral equations known asVolterra integral equations. Take the Laplace transform of the given integral equation and obtain an expression for L{(t)} in terms of the transforms L{f(t)} and L{k(t)} of the given functions f and k. The inverse transform of L{(t)} is the solution of the original integral equation.

Consider the Volterra integral equation (see Problem 21) (t) + t 0 (t )() d = sin 2t. (i) (a) Solve the integral equation (i) by using the Laplace transform. (b) By differentiating Eq. (i) twice, show that (t) satisfies the differential equation (t) + (t) = 4 sin 2t. Show also that the initial conditions are (0) = 0,  (0) = 2. (c) Solve the initial value problem in part (b), and verify that the solution is the same as the one in part (a).

In each of Problems 23 through 25: (a) Solve the given Volterra integral equation by using the Laplace transform. (b) Convert the integral equation into an initial value problem, as in Problem 22(b). (c) Solve the initial value problem in part (b), and verify that the solution is the same as the one in part (a). (t) + t 0 (t )() d = 1 2

In each of Problems 23 through 25: (a) Solve the given Volterra integral equation by using the Laplace transform. (b) Convert the integral equation into an initial value problem, as in Problem 22(b). (c) Solve the initial value problem in part (b), and verify that the solution is the same as the one in part (a). (t) t 0 (t )() d = 1 2

In each of Problems 23 through 25: (a) Solve the given Volterra integral equation by using the Laplace transform. (b) Convert the integral equation into an initial value problem, as in Problem 22(b). (c) Solve the initial value problem in part (b), and verify that the solution is the same as the one in part (a). (t) + 2 t 0 cos(t )() d = et

There are also equations, known as integro-differential equations, in which both derivatives and integrals of the unknown function appear. In each of Problems 26 through 28: (a) Solve the given integro-differential equation by using the Laplace transform. (b) By differentiating the integro-differential equation a sufficient number of times, convert it into an initial value problem. (c) Solve the initial value problem in part (b), and verify that the solution is the same as the one in part (a).  (t) + t 0 (t )() d = t, (0) = 0

There are also equations, known as integro-differential equations, in which both derivatives and integrals of the unknown function appear. In each of Problems 26 through 28: (a) Solve the given integro-differential equation by using the Laplace transform. (b) By differentiating the integro-differential equation a sufficient number of times, convert it into an initial value problem. (c) Solve the initial value problem in part (b), and verify that the solution is the same as the one in part (a).  (t) 1 2 t 0 (t )2 () d = t, (0) = 1

There are also equations, known as integro-differential equations, in which both derivatives and integrals of the unknown function appear. In each of Problems 26 through 28: (a) Solve the given integro-differential equation by using the Laplace transform. (b) By differentiating the integro-differential equation a sufficient number of times, convert it into an initial value problem. (c) Solve the initial value problem in part (b), and verify that the solution is the same as the one in part (a).  (t) + (t) = t 0 sin(t )() d, (0) = 1

The Tautochrone. A problem of interest in the history of mathematics is that of finding the tautochrone6the 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 (1629 1695) in 1673 by geometrical methods, and later by Leibniz and Jakob Bernoulli using analytical arguments. Bernoullis solution (in 1690) was one of the first occasions on which a differential equation was explicitly solved. The geometric 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: f(y) = ds dy = 1 + dx dy 2 1/2 . (i) 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 T(b) = 1 2g b 0 f(y) b y dy. (ii) y (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) = 2g T0 s ; (iii) then show that f(y) = 2g T0 y . (iv) Hint: See Problem 31 of Section 6.1. (b) Combining Eqs. (i) and (iv), show that dx dy =  2 y y , (v) where = gT2 0 /2. (c) Use the substitution y = 2 sin2 (/2) to solve Eq. (v), and show that x = ( + sin ), y = (1 cos ). (vi) Equations (vi) can be identified as parametric equations of a cycloid.Thus the tautochrone is an arc of a cycloid.