AMATH 351 Autumn 2014 Homework 5 Due: Wednesday, November 5, 2014, 10:30AM

AMATH 351 Autumn 2014
Homework 5
Due: Wednesday, November 5, 2014, 10:30AM
Show work for full credit!
1. Find the Laplace transform of the function.
s > |b|
(a) f (t) = cosh bt,
(b) f (t) = eat cos bt,
s>a
2. Laplace transform I
Use the Laplace transform to solve the initial value problem.
(a) y 00 + 3y 0 + 2y = 0,
y(0) = 1,
y 0 (0) = 0
(b) y 00 − 4y 0 + 4y = 0,
y(0) = 1,
y 0 (0) = 1
(c) y 00 + 2y 0 + 5y = 0,
y(0) = 2,
y 0 (0) = −1
(d) y 00 + 4y = 4t,
y(0) = 1,
y 0 (0) = 5
3. The Heaviside function
Sketch a graph of the given functions on the interval t ≥ 0. uc (t) represents the Heaviside
(unit step) function.
(a) g(t) = u1 (t) + 2u3 (t) − 6u4 (t)
(b) g(t) = sin(t − 3)u3 (t)
Express f (t) in




(c) f (t) =



terms of uc (t) and sketch a graph of f (t).
t,
2,
7 − t,
0,
0≤t<2
2≤t<5
5≤t<7
t≥7
1
4. The Gamma function
The Gamma function is denoted by Γ(p) and is defined by the integral:
Z ∞
Γ(p + 1) =
e−x xp dx
0
The integral converges as x → ∞ for all p. For p < 0 is it improper because the integrand
becomes unbounded as x → 0. However, the integral can be shown to converge at x = 0 for
p > −1.
(a) Show that Γ(p + 1) = pΓ(p) for p > 0.
(b) Show that Γ(1) = 1.
(c) Using parts (a) and (b), find Γ(p) for p = 2, 3, 4, 5, 6. Is there a pattern? Find the
pattern and write the formula for Γ(p) for any integer p > 0.
Since Γ(p) is also defined when p is not an integer, this function provides an extension
of the factorial function
√ to non-integer values of the independent variable. It is possible to
show that Γ(1/2) = π.
(d) Find Γ(3/2) and Γ(11/2).
2
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