4—Differential Equations 94each case explain why the result is as it should be.
Ans:(F 0 /m)[−cosω 0 t+ cosωt]/(ω^20 −ω^2 )
4.9 In the preceding problem I specified thatω 6 =ω 0 =
√
k/m. Having solved it, you know why this
condition is needed. Now take the final result of that problem, including the initial conditions, and take
the limit asω→ω 0. [What is thedefinitionof a derivative?] You did draw a graph of your result didn’t
you? Ans:
(
F 0 / 2 mω 0
)
tsinω 0 t
4.10 Show explicitly that you can write the solution Eq. (4.7) in any of several equivalent ways,
Aeiω^0 t+Be−iω^0 t=Ccosω 0 t+Dsinω 0 t=Ecos(ω 0 t+φ)
I.e., givenAandB, what areCandD, what areEandφ? Are there any restrictions in any of these
cases?
4.11 In the damped harmonic oscillator, you can have the special (critical damping) case for which
b^2 = 4kmand for whichω′= 0. Use a series expansion to take the limit of Eq. (4.10) asω′→ 0.
Also graph this solution. What would happen if you took the same limit in Eqs. (4.8) and (4.9),before
using the initial conditions?
4.12 (a) In the limiting solution for the forced oscillator, Eq. (4.16), what is the nature of the result
for small time? Expand the solution through ordert^2 and understand what you get. Be careful to be
consistent in keeping terms to the same order int.
(b) Part (a) involved lettingβbe very large, then examining the behavior for smallt. Now reverse the
order: What is the first non-vanishing order intthat you will get if you go back to Eq. (4.13), expand
that to first non-vanishing order in time, use that for the external force in Eq. (4.12), and findx(t)for
smallt. Recall that in this examplex(0) = 0andx ̇(0) = 0, so you can solve forx ̈(0)and then for
̈x ̇(0). The two behaviors are very different.
4.13 The undamped harmonic oscillator equation isd^2 x/dt^2 +ω^2 x= 0. Solve this by Frobenius series
expansion aboutt= 0.
4.14 Check the algebra in the derivation of then= 0Bessel equation. Explicitly verify that the general
expression fora 2 kin terms ofa 0 is correct, Eq. (4.22).
4.15 Work out the Frobenius series solution to the Bessel equation for then=^1 / 2 , s=−^1 / 2 case.
Graph both solutions, this one and Eq. (4.23).
4.16 Derive the Frobenius series solution to the Bessel equation for the value ofn= 1. Show that this
method doesn’t yield a second solution for this case either.
4.17 Try using a Frobenius series method ony′′+y/x^3 = 0aroundx= 0.
4.18 Solve by Frobenius seriesx^2 u′′+ 4xu′+ (x^2 + 2)u= 0. You should be able to recognize the
resulting series (after a little manipulation).