This document contains 4 questions regarding differential equations, Legendre polynomials, and Bessel functions. Question 1 asks to solve 10 differential equations in series form. Question 2 asks to prove 10 properties related to Legendre polynomials and Bessel functions. Question 3 asks to state and prove 5 properties related to Rodrigue's formula, orthogonality of Legendre polynomials, orthogonality of Bessel functions, and generating functions. Question 4 asks to express functions in terms of Legendre polynomials and Bessel functions, and prove 5 additional properties regarding Bessel functions and Legendre polynomials.

1. Question Bank Unit II Engineering Mathematics II (Section P7)
Q 1 Solve the following differential equations in series (10 marks each)
1) 2 x 2 y ′′ − xy ′ + ( x − 5) y = 0 (2007)
2) x 2 y ′′ + x( x − 1) y ′ + (1 − x) y = 0 (2008)
3) 4) xy ′′ + y ′ + x 2 y = 0 (2006)
4) ((1 − x 2 ) y ′)′ + 2 y = 0 (2008)
5) (1 − x 2 ) y ′′ − xy ′ − 4 y = 0 (using Frobenius method) (2008,2012)
Q 2 Prove the following (5 marks each)
1
2 (−1) n (2n)!
1) ∫ [ Pn ( x)] = (2007, 2008), 2) P2 n (0) = 2 n
2
(2007) ,
−1
2n + 1 2 ( n !) 2
1 1
(−1) n
∫1 f ( x) Pn ( x)dx = ∫ ( x − 1) f ( x)dx (2008)
2 n (n)
3)
− 2 n n ! −1
4) Pn′+1 ( x ) − Pn′−1 ( x ) = (2n + 1) Pn ( x ) (2008),
∞
2 −1/2
5) (1 − 2 xz − z ) = ∑ Pn ( x) z n (2008)
0
2 ′
6) J1/2 ( x) = sin x (2007), 7) J n ( x) = nJ n ( x) − xJ n +1 ( x) (2009)
πx
8) (n + 1) Pn +1) ( x) = (2n + 1) xPn ( x) − nPn −1 ( x) , (2009)
1 1
9) 5) y ′′ + y ′ + (8 − 2 ) y = 0 , in terms of Bessel’s functions (2012)
x x
10) ( x J n ( x))′ = x J n −1 ( x ) (2008,2010)
n n
Q 3 (10 marks each)
1) State and prove Rodrigue’s formula
2) State and prove the orthogonality property of Legendre polynomials
3) State and prove the orthogonality property of Bessel’s functions
5) State and prove generating function formula for Legendre polynomials (Bessel’s
functions)
Q 4 1) Express f ( x) = x 4 + 3 x3 − x 2 + 5 x − 2 in terms of Legendre polynomials (2012)
2) Express J 6 ( x ) in terms of J 0 ( x) and J1 ( x) (2009)
1 2 2
3) Show that ∫ xJ 0 ( x )dx = x ( J 0 ( x ) + J1 ( x)) + c (2011)
2 2
2
 8  4
4) Show that J 3 ( x) =  2 − 1÷J1 ( x) − J 0 ( x) (2012)
x  x
0, m ≠ n
5) Show that ∫ (1 − x ) Pm
2 ′ Pn′ dx =  2n(n + 1)

 2n + 1 , m=n
