Mathematical Tools for Physics

4—Differential Equations 116

twice with respect to time to verify that it really gives what it’s supposed to. This is a special case of some
general results, problems15.19and15.20.

4.39 A point massmmoves in one dimension under the influence of a forceFxthat has a potential energy
V(x). Recall that the relation between these is

Fx=−

dV

dx

Take the specific potential energyV(x) =−V 0 e−x

(^2) /a 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. As usual, analyze large and smalla.
4.40 Solve by Frobenius series methods
d^2 y
dx^2

+

2

x

dy

dx

+

1

x

y= 0

4.41 Find a series solution aboutx= 0fory′′+ysecx= 0, at least to a few terms.
Ans:a 0

[

1 −^12 x^2 + 0x^4 + 7201 x^6 +···

]

+a 1

[

x−^16 x^3 + 601 x^5 +···

]

4.42 Fill in the missing steps in the equations ( 36 ) to Eq. ( 39 ).

4.43 Verify the orthogonality relation Eq. ( 42 )(a) for the Legendre polynomials of order`= 0, 1 , 2 , 3.

4.44 Start with the function

(

1 − 2 xt+t^2

)− 1 / 2

. Use the binomial expansion and collect terms to get a power
series int. The coefficients in this series are functions ofx. Carry this out at least to the coefficient oft^3 and
show that the coefficients are Legendre polynomials. This is called the generating function for the∑ P’s. It is ∞ 0 P(x)t

`