Chapter 1
Homework 1
1.1 Verify that the solution of the initial value problem
mx00 + kx = 0,
x(t0 ) = x0 ,
x0 (t0 ) = v0
is given by
p
p
v0
sin( k/m[t − t0 ]).
x(t) = x0 cos( k/m[t − t0 ]) + p
k/m
A calculation gives
p
p
p
x0 (t) = −x0 k/m sin( k/m[t − t0 ]) + v0 cos( k/m[t − t0 ]),
and
p
p
p
x00 (t) = −x0 (k/m) cos( k/m[t − t0 ]) − v0 k/m sin( k/m[t − t0 ]),
so
mx00 + kx =
p
√
p
−x0 k cos( k/m[t − t0 ]) − v0 mk sin( k/m[t − t0 ])
p
p
√
+x0 k cos( k/m[t − t0 ]) + v0 km sin( k/m[t − t0 ]) = 0.
We also have
x0 (t0 ) = v0 ,
x(t0 ) = x0 ,
as desired.
1.2 a) Find the general solution of
x00 +
a
k
x = cos(ωt)
m
m
1
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CHAPTER 1. HOMEWORK 1
b) Find the general solution of
x00 +
a1
a2
k
x=
cos(ωt) + sin(ωt).
m
m
m
a) Look for a solution of the form
xp (t) = b cos(ωt).
We get
x0p (t) = −bω sin(ωt),
and
x00p (t) = −bω 2 cos(ωt),
so we need
b[−ω 2 +
or
b=
a
k
]= ,
m
m
1
a
.
m [k/m − ω 2 ]
b) First notice that a particular solution is obtained by adding the cases we
know,
xp (t) = b1 sin(ωt) + b2 cos(ωt),
with
b1 =
a1
1
,
m [k/m − ω 2 ]
b2 =
a2
1
.
m [k/m − ω 2 ]
The general solution is obtained by adding a general solution of the homogeneous
equation
x = xp (t) + c1 cos(ωt) + c2 sin(ωt).
1.3 Show that
sin(a) cos(t) + cos(a) sin(t) = sin(a + t),
cos(a) cos(t) − sin(a) sin(t) = cos(a + t).
One checks easily that sin(a + t) and cos(a + t) are each solutions of the linear
equation
x00 + x = 0.
Thus each one can be written as a linear combination of the basis solutions sin(t)
and cos(t),
x1 (t) = sin(a + t) = b1 cos(t) + b2 sin(t).
x2 (t) = cos(a + t) = c1 cos(t) + c2 sin(t).
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Evaluation at t = 0 gives
x01 (0) = cos(a) = b2 ,
x1 (0) = sin(a) = b1 ,
and
x02 (0) = − sin(a) = c2 ,
x2 (0) = cos(a) = c1 ,
as desired.
1.4 Show that if the function
x(t) = c cos(ω0 t) + d sin(ω0 t) +
a
sin(ωt),
− ω2]
m[ω02
a 6= 0,
ω 6= ω0
has period 2π/ω, then c = d = 0 unless ω0 /ω is an integer.
For notational simplicity let
x1 (t) = c cos(ω0 t) + d sin(ω0 t),
x2 (t) =
a
sin(ωt).
− ω2]
m[ω02
The assumption of periodicity means that
x(t + 2π/ω) = x(t).
Since the function x2 (t) is periodic with period 2π/ω we conclude that
x1 (t + 2π/ω) = x(t + 2π/ω) − x2 (t + 2π/ω) = x(t) − x2 (t) = x1 (t),
so x1 (t) is periodic with period 2π/ω.
By trig identities (or the uniqueness of solutions to x00 + ω02 x = 0) we have
x1 (t) = A cos(ω0 t + φ),
with
A=
√
c2 + d 2 ,
d
tan(φ) = − .
c
The periodicity now implies
cos(ω0 t + φ) = cos(ω0 [t + 2π/ω] + φ) = cos(ω0 t + 2πω0 /ω + φ),
or letting s = ω0 t + φ,
cos(s) = cos(s + 2πω0 /ω).
The cosine function has least period 2π, so
2πω0 /ω = 2nπ
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CHAPTER 1. HOMEWORK 1
for some integer n.
1.5 Considering equations 1.1. and 1.2 in the text, when will the concentration
of toxin in lake 2 be greatest?
A reasonable guess is that the concentration of toxin in lake 2 be greatest at
the end of day 1. The formula
τ1 (t) = c0 v1 [1 − e−ti1 /v1 ]
shows that the concentration of toxin in lake 1 is positive and increasing during
the first day, 0 ≤ t ≤ 1. The formula for τ2 is
Z
i2 t o2 (s−t)/v2
e
τ1 (s) ds.
τ2 (t) =
v1 0
It helps to make the substitution u = t − s, getting
Z
i2 t −o2 u/v2
τ2 (t) =
e
τ1 (t − u) du.
v1 0
Suppose that t1 < t2 . Then
Z
i2 t1 −o2 u/v2
[τ1 (t2 − u) − τ1 (t1 − u)] du
e
τ2 (t2 ) − τ2 (t1 ) =
v1 0
Z
i2 t2 −o2 u/v2
+
τ1 (t2 − u) du.
e
v 1 t1
Since τ1 is increasing and t2 > t1 , the function τ1 (t2 −u)−τ1 (t1 −u) is nonnegative.
The second integral on the right is positive, so
τ2 (t2 ) − τ2 (t1 ) ≥ 0,
or τ2 increases with time during day 1.
1.8 a) Using the chain rule, the equation
mx0 x00 + kxx0 = 0
may also be written
1d
[m(x0 )2 + kx2 ] = 0.
2 dt
Integration gives
[m(x0 )2 + kx2 ](t) − [m(x0 )2 + kx2 ](0) = 0,
5
or E(t) = E(0).
b) The equation has the form
ax2 + b(x0 )2 = c,
which is the equation of an ellipse in the (x, x0 ) plane.
c) Starting with
mx00 + cx0 + kx = 0,
m, c, k > 0,
begin by writing this as
mx00 x0 + c(x0 )2 + kxx0 = 0.
This can be rewritten as
1d
[m(x0 )2 + kx2 ] = −c(x0 )2 .
2 dt
Integration gives
E(t) − E(0) = −
or
E(t) = E(0) −
Z
Z
t
c(x0 (s))2 ds,
0
t
c(x0 (s))2 ds,
0
which says that the energy is a nonincreasing function of time.
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CHAPTER 1. HOMEWORK 1
Bibliography
[1] A. Peressini and F. Sullivan and J. Uhl. The Mathematics of Nonlinear
Programming. Springer, 1988.
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