4—Differential Equations 1114.8 For the undamped harmonic oscillator apply an extra oscillating force so that the equation to solve is
md^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 acosωt
from a combination ofxand its second derivative.
Add these and apply the initial conditions that at timet= 0the mass is at rest at the origin. Be sure to check
your results for plausibility: 0) dimensions; 1)ω= 0; 2)ω→∞; 3)tsmall (not zero). In each 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?
4.10 Show explicitly that you can write the solution Eq. ( 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 case for whichb^2 = 4kmand for whichω′= 0.
Use a series expansion to take the limit of Eq. ( 10 ) asω′→ 0. Also graph this solution. What would happen if
you took the same limit in Eqs. ( 8 ) and ( 9 ),beforeusing the initial conditions?
4.12 In the limiting solution for the forced oscillator, Eq. ( 14 ), 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.
4.13 The undamped harmonic oscillator equation isd^2 x/dt^2 +ω^2 x= 0. Solve this by series expansion about
t= 0.