4—Differential Equations 1134.24 Apply the Green’s function method for the forceF 0
(
1 −e−βt)
on the harmonic oscillator without damping.
Verify that it agrees with the previously derived result, Eq. ( 13 ). They should match in a special case.
4.25 An undamped harmonic oscillator with natural frequencyω 0 is at rest for timet < 0. Starting at time zero
there is an added forceF 0 sinω 0 t. Use Green’s functions to find the motion for timet > 0. Analyze the solution
for both small and large time, determining if your results make sense. Compare the solution to problem 9.
4.26 Derive the Green’s function analogous to Eq. ( 21 ) for the case that the harmonic oscillator is damped.
4.27 Radioactive processes have the property that the rate of decay of nuclei is proportional to the number of
nuclei present. That translates into the differential equationdN/dt=−λN, whereλis a constant depending on
the nucleus. At timet= 0there areN 0 nuclei; how many are present at timetlater? The half-life is the time
in which one-half of the nuclei decay; what is that in terms ofλ? Ans:ln 2/λ
4.28 In the preceding problem, suppose that the result of the decay is another nucleus (the “daughter”) that is
itself radioactive with its own decay constantλ 2. Call the first one aboveλ 1. Write the differential equation for
the time-derivative of the number,N 2 of this nucleus. You note thatN 2 will change for two reasons, so in time
dtthe quantitydN 2 has two contributions, one is the decrease because of the radioactivity of the daughter, the
other an increase due to the decay of the parent. Set up the differential equation forN 2 and you will be able to
use the result of the previous problem as input to this and then to solve the resulting differential equation for the
number of daughter nuclei as a function of time. Assume that you started with none,N 2 (0) = 0.
(b) The “activity” is the total number ofalltypes of decays per time. Compute the activity and graph it. For
the plot, assume thatλ 1 is substantially smaller thanλ 2 and plot the total activity as a function of time. Then
examine the reverse case,λ 1 λ 2
Ans: (b)N 0 λ 1
[
(2λ 2 −λ 1 )e−λ^1 t−λ 2 e−λ^2 t]
/(λ 2 −λ 1 )4.29 The “snowplow problem” was made famous by Ralph Agnew: A snowplow starts out at 12:00 Noon in a
heavy and steady snowstorm. In the first hour it goes 2 miles; in the second hour it goes 1 mile. When did the
snowstorm start? Ans: 11:23
4.30 Verify that the equations ( 33 ) really do satisfy the original differential equations.