Mast 330 / Math 370 Sec A Final Exam December 2002

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Mast 330 / Math 370 Sec A Final Exam December 2002
Professor:
Instructions:
Richard Hall
Please answer all 5 questions, which carry equal marks.
Explain your working carefully. Calculators of any type are
permitted. [Lined booklets]
1. Solve the following differential equations by first finding the general solution,
and then the particular solution satisfying the given condition:
(a)
y ′ = t 2 cot(y), t ≥ 0, y(1) = π 3 .
(b)
(3t 2 − t 3 − ty + y)dt + tdy =0, y(1) = 1.
2. A group of rabbits enters a very prosperous period. In a continuous model,
the number of rabbits N(t) at time t (in months) satisfies the differential
equation
dN
dt = rN + λt2 , r, λ > 0.
(a) Find the general solution.
(b) For the case r = 1 2
, λ =1, find the particular solution satisfying N(0) =
20. What is the population N(5) in five months
3. Find the general solution to the differential equation
y ′′ − 2y ′ + y =
et
1+t 2 .

Mast 330 / Math 370 Sec A Final Exam December 2002 page 2 of 3
4. Consider the differential equation representing forced vibrations of a springmass
system with friction, given by
x ′′ + x ′ + 5 4 x = cos(ωt),
where x(t) is the position of the mass at time t, and ω is a positive parameter.
(a) Find the general solution.
(b) Find C(w), the amplitude of the steady-state (large-time) solution.
(c) Sketch C(ω), the ‘resonance curve’, and find for what value ˆω of ω C(ω)
is maximum.
5. By using Laplace transforms or otherwise solve the following initial-value problem
in which a linear system, initially at rest, receives an impulse at t =1:
y ′′ − y =3δ(t − 1), y(0) = y ′ (0) = 0.
Verify that your answer to this problem is indeed a solution to the given differential
equation.

Mast 330 / Math 370 Sec A Final Exam December 2002 page 3 of 3
Table of Laplace Transforms
f(t) =L −1 {F (s)}
1
F (s) =L{f(t)} = ∫ ∞
0 e−st f(t)dt
1
s , s > 0
e at 1
s−a , s>a
t n , n integer > 0
n!
s n+1 , s > 0
t p ,
p > −1
sin(bt)
cos(bt)
Γ(p+1)
,s>0
s p+1
b
s 2 +b 2 , s > 0
s
s 2 +b 2 , s > 0
sinh(bt)
cosh(bt)
e at sin(bt)
e at cos(bt)
e at f(t)
b
s 2 −b 2 , s > |b|
s
s 2 −b 2 , s > |b|
b
(s−a) 2 +b 2 , s>a
s−a
(s−a) 2 +b 2 , s>a
F (s − a), s > a
(−t) n f(t), n integer > 0 F (n) (s)
f (n) (t) s n F (s) − s n−1 f(0) − ...− f (n−1) (0)
f(ct)
u c (t)
1
c F ( )
s
c , c > 0
e −cs
s ,c≥ 0
u c (t)f(t − c) e −cs F (s), c ≥ 0
δ(t − c) e −cs , c ≥ 0
∫ t
0
f(t − τ)g(τ)dτ
F
(s)G(s)