(a) Laguerre’s differential equation is known to possess

Chapter 7, Problem 61E

(choose chapter or problem)

(a) Laguerre’s differential equation

\(t y^{\prime \prime}+(1-t) y^{\prime}+n y=0)

is known to possess polynomial solutions when n is a nonnegative integer. These solutions are naturally called Laguerre polynomials and are denoted by \(L_{n}(t)\). Find \(y=L_{n}(t)\), for n  0, 1, 2, 3, 4 if it is for n = 0, 1, 2, 3, 4 if it is known that \(L_{n}(0)=1).

(b) Show that

\(\mathscr{L}\left\{\frac{e^{t}}{n !} \frac{d^{n}}{d t^{n}} t^{n} e^{-t}\right\}=Y(s)\).

where \(Y(s)=\mathscr{L}\{y\}\), and \(y=L_{n}(t)\) is a polynomial solution of the DE in part (a). Conclude that

\(L_{n}(t)=\frac{e^{t}}{n !} \frac{d^{n}}{d t^{n}} t^{n} e^{-t}\), n = 0, 1, 2, . . .

This last relation for generating the Laguerre polynomials is the analogue of Rodrigues’ formula for the Legendre polynomials. See (33) in Section 6.4.

Text Transcription:

t y^prime prime+(1-t) y^prime +n y=0

L_n t

y=L_n t

L_n 0=1

mathscr L {{e^t}/n ! d^n/d t^n} t^n e^-t}=Y(s)

Y(s)=mathscr L y

L_n(t)={e^t}/n ! {d^n/d t^n} t^n e^-t