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THE ASSOCIATED LEGENDRE FUNCTIONS: m dm P" (x) = (1 – x²) dxm P(x) Example: Make the change of variable x= cos 8 in Sing do Si79 To find:

THE ASSOCIATED LEGENDRE FUNCTIONS: m dm P" (x) = (1 – x²) dxm P(x) Example: Make the change of variable x= cos 8 in Sing do Si79 To find:

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Explain this proof in very boring details please

THE ASSOCIATED LEGENDRE FUNCTIONS: dm P.(x) m P™ (x) = (1 – x²). dxm Example: Make the change of variable x= cos 8 in m To find: Si79 (1 – x²) y" – 2xy' + m2 - 1) – y = 0 with m² s1? 1- x2 Somtion:- dz cos8 > sind 00 dy dy dy (- sin@) %3D dy - sing dy oy sin'g + cos'8 = 1 → sin'9 = Vx² %3D brom (2) into c1): dy do dy %3D | - VT-x² %3D doc dy %3D %3D de de d'y dx = - - z' d'y
Transcribed Image Text:THE ASSOCIATED LEGENDRE FUNCTIONS: dm P.(x) m P™ (x) = (1 – x²). dxm Example: Make the change of variable x= cos 8 in m To find: Si79 (1 – x²) y" – 2xy' + m2 - 1) – y = 0 with m² s1? 1- x2 Somtion:- dz cos8 > sind 00 dy dy dy (- sin@) %3D dy - sing dy oy sin'g + cos'8 = 1 → sin'9 = Vx² %3D brom (2) into c1): dy do dy %3D | - VT-x² %3D doc dy %3D %3D de de d'y dx = - - z' d'y
dy %D xp OP For a given eguation ;- (Sin 8 dy ]y= 0 Sinig Sind dg we get : (Sin8 d, ]y= o cos 8 d)+[ eceai) - Sin'g Sin e cos 8 dy +[ec&+1) - Sin g do Sin? g From (3) and (5) into (6) d'y dy dy + [ ecepi) - ]y = 0 Jy =0 | d'y dy dy ece t)- y] = 0 %3D (1- x²) dy dy - 2 x + [ ec& +1) - Jy=0 dx?
Transcribed Image Text:dy %D xp OP For a given eguation ;- (Sin 8 dy ]y= 0 Sinig Sind dg we get : (Sin8 d, ]y= o cos 8 d)+[ eceai) - Sin'g Sin e cos 8 dy +[ec&+1) - Sin g do Sin? g From (3) and (5) into (6) d'y dy dy + [ ecepi) - ]y = 0 Jy =0 | d'y dy dy ece t)- y] = 0 %3D (1- x²) dy dy - 2 x + [ ec& +1) - Jy=0 dx?
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