Mathematical Tools for Physics

4—Differential Equations 115

4.35 For the damped harmonic oscillator apply an extra oscillating force so that the equation to solve is

m

d^2 x

dt^2

=−b

dx

dt

−kx+Fext(t)

where the external force isFext(t) =F 0 eiωt.
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 aneiωt
from a combination ofxand its first two derivatives.
This problem is easier to solve than the one usingcosωt, and at the end, to get the solution for the cosine case,
all you have to do is to take the real part of your result.

4.36 You can solve the circuit equation Eq. ( 24 ) more than one way. Solve it by the methods used earlier in this
chapter.

4.37 For a second order differential equation you can pick the position and the velocity any way that you want,
and the equation then determines the acceleration. Differentiate the equation and you find that the third derivative
is determined too.
d^2 x
dt^2

=−

b

m

dx

dt

−

k

m

x implies

d^3 x

dt^3

=−

b

m

d^2 x

dt^2

−

k

m

dx

dt

Assume the initial position is zero,x(0) = 0and the initial velocity isvx(0) =v 0 ; determine the second derivative
at time zero. Now determine the third derivative at time zero. Now differentiate the above equation again and
determine the fourth derivative at time zero.
From this, write down the first five terms of the power series expansion ofx(t)aboutt= 0.
Compare this result to the power series expansion of Eq. ( 10 ) to this order.

4.38 Use the techniques of section4.5, start from the equationmd^2 x/dt^2 = Fx(t)withno spring force or
damping. Find the Green’s function for this problem, that is, what is the response of the mass to a small kick
over a small time interval (the analog of Eq. ( 21 ))? Develop the analog of Eq. ( 23 ) for this case. Apply your
result to the special case thatFx(t) =F 0 , a constant for timet > 0.
(b) You know that the solution of this differential equation involves two integrals ofFx(t)with respect to time,
so how can this single integral do the same thing? Differentiate this Green’s function solution (for arbitraryFx)