Question

The Legendre equation of order p is, a) Find the associated Euler equation and the characteristic equation for x = 1. b) Find the first three nonzero terms in one of the power series solution in powers of r -1 for x-10 Hint: Write 1 + x 2 + (2-1) and x = 1 + (x-1). Alternatively, make the change of variable x- 1-t and determine the series solution in powers of t.

Answer 1

a) can be written as

1 a2 2y p(p+ 1)y= 0

Рp+ 1) у— 0 (1-т?) 2т у" + (1- г?)

$x=1$ is a singular point of the differential equation.

Since $x=1$ is a pole of order 1, it is a regular singular point.

b)For  
Рp+ 1) у— 0 (1-т?) 2т у" + (1- г?)
put

Then the equation transforms to

2(1 t Pp+1) y" t(t2) y=0 + t(t2)

inf inf y(t) ant n-2"2 n-0 n-2 inf inf (t)-na, -n-1an-1 2 n 1 inf n 2 y"(t) n(n -1)a,t"2 n-2

Substitute in the above equation to solve it.

inf inf int n(n-1)a,n-2 2(1t) + t(t2)(n-1)a-t-2 P(p+1) an-2t-2 t(t2) n-2 n-2 n=2

Solution upto first three terms is

2(1t) РP+1) t(t2) azt0 4(1t) Рp+1) t(t2) ait12a4t 2a2 t(t 2) ao6a3t azt t(t 2) p(p1) 6(1t - a3t t(t2) ()

2a2t(t2) +2(1t)a1 p(p1)ao6a3t (t1)4(1+ t)a2t p(p+ 1)a1t12a4t + 6(1+ t)a3t + p(p1)a2t 0

Comapring coefficients on both sides,

2a1tpp 1)ao 0 2a1t4a2tp(p + 1)ait6a3t12a4t 2a2t26a3t24a2t 6a3tpp+ 1)a2t2 = 0

Using the above three equations solve for