4—Differential Equations 110Problems4.1 If the equilibrium positionx= 0for Eq. ( 4 ) is unstable instead of stable, this reverses the sign in front of
k. Solve the problem that led to Eq. ( 10 ) under these circumstances. That is, the initial conditions arex(0) = 0
andvx(0) =v 0. What is the small time and what is the large time behavior?
4.2 In the damped harmonic oscillator problem, Eq. ( 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 condition
x(0) = 0andvx(0) =v 0 and describe the resulting behavior.
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
dxTake the specific potential energyV(x) =−V 0 a^2 /(a^2 +x^2 ), whereV 0 is positive. SketchV. Write the equation
Fx=max. There is an equilibrium point atx= 0, and if the motion is over only small distances you can do a
power series expansion ofFx aboutx= 0. What is the differential equation now? Keep only 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.
(b) How does the graph ofV change as you varyafrom small to large values and how does this same change in
aaffect the behavior of your solution? Ans:ω=
√
2 V 0 /ma^24.4 The same as the preceding problem except that the potential energy function isV(x) = +V 0 a^2 /(a^2 +x^2 ).
4.5 For the case of the undamped harmonic oscillator and the force Eq. ( 12 ), 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. ( 13 ) to see if they agree where they should agree.
4.6 Check the dimensions in the result for the forced oscillator, Eq. ( 13 ).
4.7 Fill in the missing steps in the derivation of Eq. ( 13 ).