Mathematical Methods in Physics - Legender Differential Equations

Layeste dffntia kayation.
Iae diffevential quation s )

(-&)
whee M is el Mnbey is calet lrqondre difgoenhál
equation. bis equa tion is f lonsderable impeme n
aypled wacterais , putulaay tn bougday vale prablrs
iovdlviag seuial tonlguatons. Tauga values
inkgad
soed n eies
be
Aso eaulon O Can


ayinding ts
han ihe one i

aacend Bing powts
et y assume




(k-) a, aK-7

k-y-2
(K-v)CK-7-1) ay


(-)k-1) CK-y-)a, exK--2
+ nn4) ar gk-yo
26-) CK-y-) ay k-y-2 Z(h-) CK-)ay K-y

,RK-7) ay

+




(k-+) (k--)


Tbis is pguaton is ralld tnhicial e4yahion Mow the
Coeffatnts vaubus oun o can be eduasvd to Xo.
taua king
CY=0) in ) to xero.
n(n+)- (k-o) (K-0+) ay = D

n(n+) - K(K+) ao = 0

ao cannot
of the
be kuo,.Sine leading oeffiuent
ssumed solntion
n(n+) - k(K+)


(Mtk) (n-k) + M-kK) = 0
(M-*) [n+K+)
n+ k+l =0




éayating the co éfuent lbwut poua
to (T=1)

, ntn+)-(K-1) (*-1+) a, -o
n(n)-(k-)k a,
tom n(y+) - K(K-) cawnot be equal to tuo wcth

should be eaual to Kuo


the coeffiaènt. of K-y to un se qet the
Eauating
YetuYomee velation Y=Y-2)



(A-Y+2)(k-y+1) ay-8 + nen+1) -(k-1)(k-y+1) Qy =D
(k-y+2) (k-i+)
n(n+)-(K-y)(K-+)
k=n
(n-+2) (n-7+1)
nn41)- (n-y)(n-Y4|)
(n-Y+a) (n-y41)


(n-1+2) (n-r)

-+a) (n-741)
(an- +)

also ony even