Lecture Notes in Differential Equations - Bruce E. Shapiro

270 LESSON 28. SERIES SOLUTIONS
Figure 28.2: Several Legendre Polynomials. P 1 (Bold); P 2 (Bold, Dotted);
P 3 (Bold, Dashed); P 4 (Thin, Dotted); P 5 (Thin, Dashed).
1.
0.5
0.
0.5
1.
1. 0.5 0. 0.5 1.
In general, the solutions of the nth equation (28.114) will give a polynomial
of order n, called the Legendre polynomial P n (t).
To solve equation (28.114) we substitute
y =
y ′ =
y ′′ =
∞∑
a k t k (28.115)
k=0
∞∑
ka k t k−1 (28.116)
k=0
∞∑
k(k − 1)a k t k−2 (28.117)
k=0