Mathematical Tools for Physics

4—Differential Equations 107

4.9 Legendre’s Equation
This equation and its solutions appear when you solve electric and gravitational potential problems in spherical
coordinates [problem9.20]. They appear when you study Gauss’s method of numerical integration [Eq. (11.26)]
and they appear when you analyze orthogonal functions [problem6.7]. Because it shows up so often it is worth
the time to go through the details in solving this equation.

[

(1−x^2 )y′

]′

+Cy= 0, or (1−x^2 )y′′− 2 xy′+Cy= 0 (36)

Assume a Frobenius solutions aboutx= 0

y=

∑∞

0

akxk+s

and substitute into ( 36 ).

(1−x^2 )

∑∞

0

ak(k+s)(k+s−1)xk+s−^2 − 2 x

∑∞

0

ak(k+s)xk+s−^1 +C

∑∞

0

akak+s= 0

∑∞

0

ak(k+s)(k+s−1)xk+s−^2 +

∑∞

0

ak

[

−2(k+s)−(k+s)(k+s−1)

]

xk+s+C

∑∞

0

akak+s= 0

∑∞

n=− 2

an+2(n+s+ 2)(n+s+ 1)xn+s−

∑∞

n=0

an

[

(n+s)^2 + (n+s)

]

xn+s+C

∑∞

n=0

anxn+s= 0

In the last equation I did the usual substitutionk=n+ 2for the first sum andk=nfor the rest. That makes
the exponents match across the equation. In the process, I simplified some of the algebraic expressions.
The indicial equation comes from then=− 2 term, which appears only once.

a 0 s(s−1) = 0, so s= 0, 1

Now set the coefficient ofxn+sto zero, and solve foran+2in terms ofan.

an+2=an

(n+s)(n+s+ 1)−C

(n+s+ 2)(n+s+ 1)

(37)