Question 2: Let f possess a continuous nth-order derivative on the interval [—1, 1]. Use Rodrigues' formula and integration by parts to show that 1 L' f(x)P, (x)dx = (1) m² | (x² - f(x)Pn(x)dx (x²-1)" fm (x)dx. ... Question 3: If the functions f₁, f2, fm (not necessarily polynomials) are orthogonal, that is, (fi, fj) = 0 for ij, and if F= c1fi+C2f2+...+cmfm Show that Ck= (F, fk) ||fk||2

Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
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Please siiir please sollllllvvve these 2 questions , thankssss sirr
Question 2: Let f possess a continuous nth-order derivative on the interval [—1, 1]. Use Rodrigues'
formula and integration by parts to show that
1
L' f(x)P, (x)dx = (1) m² | (x² -
f(x)Pn(x)dx
(x²-1)" fm (x)dx.
...
Question 3: If the functions f₁, f2, fm (not necessarily polynomials) are orthogonal, that is, (fi, fj) = 0
for ij, and if
F= c1fi+C2f2+...+cmfm
Show that
Ck=
(F, fk)
||fk||2
Transcribed Image Text:Question 2: Let f possess a continuous nth-order derivative on the interval [—1, 1]. Use Rodrigues' formula and integration by parts to show that 1 L' f(x)P, (x)dx = (1) m² | (x² - f(x)Pn(x)dx (x²-1)" fm (x)dx. ... Question 3: If the functions f₁, f2, fm (not necessarily polynomials) are orthogonal, that is, (fi, fj) = 0 for ij, and if F= c1fi+C2f2+...+cmfm Show that Ck= (F, fk) ||fk||2
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