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18.034 Problem Set #6
(modified on April 8, 2009)
Due by Friday, April 10, 2009, by NOON
1. (Laplace transform of tr). The Gamma function is defined by the integral
∞
Γ(r + 1) = e−ttrdt.
0
(a) Show that the improper integral converges for all r > −1.
(b) Show that Γ(r + 1) = rΓ(r) for r > 0. Show that Γ(1) = 1, Γ(1/2) = √π.
(c) For r > −1 show that L[tr] = Γ(r + 1)/sr+1
, s > 0
2. (a) Find the solution of the initial value problem
y�� + ω2
y = h(t) sin t − h(t − c)sin t, y(0) = y�(0) = 0,
where c > 0 is a constant and ω2 = 1.
(b) Show that y(0) = y�(0) = y��(0) = 0. Show that y and y� are continuous at t = c.
(c) Show that y��(c+) − y�� (c−) = − sin c, which is 0 if and only if c = nπ for n an integer. This
behavior is explained by that the function h(t − c) sin t is continuous at c if and only if sin c = 0.
3. Find the Laplace transform of a full rectified wave∗ f(t) = | sin t|.
4. Find the solution of the initial value problem
y�� + y = Aδ(t − c), y(0) = a, y�(0) = b,
where A, a, b, c are constant and c > 0.
(b) Show that y(t) = 0 for t ≥ c if and only if y(c) = 0 and
A = a2 + b2
, a sin c ≥ b cos c; A = − a2 + b2
, a sin c ≤ b cos c.
To interpret, these amplitudes A and locations of impulse c cancel the oscillation.
5. (The Volterra integral equation). Consider the integral equation
t 1
y(t) + (t − s)y(s)ds = −4 sin 2t.
0
(a) Show that the above integral equation is equivalent to the initial value problem
1
y�� + y = sin 2t, y(0) = 0, y�(0) = −2.
(b) Solve the integral equation by using the Laplace transform.
∗It describes a direct current.
1
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