1540470959-Boundary_Value_Problems_and_Partial_Differential_Equations__Powers

338 Chapter 5 Higher Dimensions and Other Coordinates

P 0 (x)= 1

P 1 (x)=x

P 2 (x)=( 3 x^2 − 1 )/ 2

P 3 (x)=( 5 x^3 − 3 x)/ 2

P 4 (x)=( 35 x^4 − 30 x^2 + 3 )/ 8

Table 3 Legendre polynomials

Figure 12 Graphs of the first five Legendre polynomials.

index are also zero. Hence, one of the solutions of
(
1 −x^2

)

y′′− 2 xy′+ 12 y= 0

is the polynomiala 1 (x− 5 x^3 / 3 ).Theothersolutionisanevenfunctionun-
bounded at bothx=±1.
Now we see that the boundedness conditions can be satisfied only ifμ^2 is
one of the numbers 0, 2 , 6 ,...,n(n+ 1 ),.... In such a case, one solution of
the differential equation is a polynomial (naturally bounded atx=±1). When
normalized by the conditiony( 1 )=1, these are calledLegendre polynomials,
writtenPn(x). Table 3 provides the first five Legendre polynomials. Figure 12
shows their graphs.