4—Differential Equations 93Problems4.1 If the equilibrium positionx= 0for Eq. (4.4) is unstable instead of stable, this reverses the sign in
front ofk. Solve the problem that led to Eq. (4.10) under these circumstances. That is, the damping
constant isbas before, and the initial conditions arex(0) = 0andvx(0) =v 0. What is the small time
and what is the large time behavior?
Ans:
(
2 mv 0 /
√
b^2 + 4km
)
e−bt/^2 msinh
(√
b^2 / 4 m+k/mt
)
4.2 In the damped harmonic oscillator problem, Eq. (4.4), suppose that the damping term is ananti-
damping term. It has the sign opposite to the one that I used (+bdx/dt). Solve the problem with the
initial conditionx(0) = 0andvx(0) =v 0 and describe the resulting behavior.
Ans:
(
2 mv 0 /
√
4 km−b^2
)
ebt/^2 msin
(√
4 km−b^2 t/ 2 m
)
4.3 A point massmmoves in one dimension under the influence of a forceFxthat has a potential
energyV(x). Recall that the relation between these is
Fx=−
dV
dx
Take the specific potential energyV(x) =−V 0 a^2 /(a^2 +x^2 ), whereV 0 is positive. SketchV. Write
the equationFx=max. There is an equilibrium point atx= 0, and if the motion is over only small
distances you can do a power series expansion ofFxaboutx= 0. What is the differential equation
now? Keep just the lowest order non-vanishing term in the expansion for the force and solve that
equation subject to the initial conditions that at timet= 0,x(0) =x 0 andvx(0) = 0.
How does the graph ofV change as you varyafrom small to large values and how does this same
change inaaffect the behavior of your solution? Ans:ω=
√
2 V 0 /ma^2
4.4 The same as the preceding problem except that the potential energy function is+V 0 a^2 /(a^2 +x^2 ).
Ans:x(t) =x 0 cosh
(√
2 V 0 /ma^2 t
)
(|x|< a/ 4 or so, depending on the accuracy you want.)
4.5 For the case of the undamped harmonic oscillator and the force Eq. (4.13), start from the beginning
and derive the solution subject to the initial conditions that the initial position is zero and the initial
velocity is zero. At the end, compare your result to the result of Eq. (4.15) to see if they agree where
they should agree.
4.6 Check the dimensions in the result for the forced oscillator, Eq. (4.15).
4.7 Fill in the missing steps in the derivation of Eq. (4.15).
4.8For the undamped harmonic oscillator apply an extra oscillating force so that the equation to solve
is
m
d^2 x
dt^2
=−kx+Fext(t)
where the external force isFext(t) =F 0 cosωt. Assume thatω 6 =ω 0 =
√
k/m.
Find the general solution to the homogeneous part of this problem.
Find a solution for the inhomogeneous case. You can readily guess what sort of function will give you