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Ordinary Differential Equations

The document provides problems involving ordinary differential equations, Legendre polynomials, and Bessel functions. It asks the reader to: 1) Find the general and series solutions to various linear differential equations. 2) Prove various identities relating Legendre polynomials and their derivatives. 3) Evaluate integrals involving Legendre polynomials. 4) Prove identities relating Bessel functions and their derivatives.

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50 views1 page

Ordinary Differential Equations

The document provides problems involving ordinary differential equations, Legendre polynomials, and Bessel functions. It asks the reader to: 1) Find the general and series solutions to various linear differential equations. 2) Prove various identities relating Legendre polynomials and their derivatives. 3) Evaluate integrals involving Legendre polynomials. 4) Prove identities relating Bessel functions and their derivatives.

Uploaded by

samir
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
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Download as PDF, TXT or read online on Scribd
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ORDINARY DIFFERENTIAL EQUATIONS

1. Find the general solution of the following linear differential equations:


a. 2𝑥 2 𝑦 ′′ + 3𝑥𝑦 ′ − 15𝑦 = 0.
b. 𝑥 2 𝑦 ′′ − 7𝑥𝑦 ′ + 16𝑦 = 0.
c. 𝑥 2 𝑦 ′′ + 3𝑥𝑦 ′ + 4𝑦 = 0.
d. 3(𝑥 + 6)2 𝑦 ′′ + 25(𝑥 + 6)𝑦 ′ − 16𝑦 = 0 in the interval not containing 𝑥 = −6.

2. Find the series solution of the following linear differential equations:


a. 𝑦 ′′ + 𝑦 = 0 about 𝑥0 = 0
b. 𝑦 ′′ − 𝑥𝑦 = 0 about 𝑥0 = 0
c. 𝑦 ′′ − 𝑥𝑦 = 0 about 𝑥0 = −2
d. (𝑥 2 + 1)𝑦 ′′ − 4𝑥𝑦 ′ + 6𝑦 = 0 about 𝑥0 = 0
e. 𝑦 ′′ − 2𝑥𝑦 ′ + 𝑦 = 0 about 𝑥0 = 0
f. 𝑦 ′′ − 𝑥𝑦 ′ + 𝑦 = 0
g. 𝑦 ′′ + 𝑥 2 𝑦 = 0 with 𝑦(0) = 0, 𝑦 ′ (0) = 1
h. 𝑥 2 𝑦 ′′ + 𝑥𝑦 ′ + 𝑥 2 𝑦 = 0 with 𝑦(0) = 1, 𝑦 ′ (0) = 0

3. If 𝑃𝑛 (𝑥) denotes legendre polynomial of order n, prove the followings:


′ ( ) ′ ( )
a. 2𝑥𝑃𝑛′ (𝑥 ) + 𝑃𝑛 (𝑥 ) = 𝑃𝑛+1 𝑥 + 𝑃𝑛−1 𝑥 ,
′ ( ) ′
b. 𝑃𝑛+1 𝑥 = (𝑛 + 1)𝑃𝑛 (𝑥 ) + 𝑥𝑃𝑛 (𝑥)
′ ( )
c. 𝑃𝑛−1 𝑥 = −𝑛𝑃𝑛 (𝑥 ) + 𝑥𝑃𝑛′ (𝑥)
d. (1 − 𝑥 2 )𝑃𝑛′ (𝑥 ) = 𝑛𝑃𝑛−1 (𝑥 ) − 𝑛𝑥𝑃𝑛 (𝑥)
e. (1 − 𝑥 2 )𝑃𝑛′ (𝑥 ) = (𝑛 + 1)𝑥𝑃𝑛 (𝑥 ) − (𝑛 + 1)𝑃𝑛+1 (𝑥 )
f. Evaluate the following integrals:
1
i. ∫−1 𝑥𝑃𝑛−1 (𝑥 )𝑃𝑛 (𝑥 )𝑑𝑥
1
ii. ∫−1 𝑥𝑃𝑛 (𝑥 )𝑑𝑥

4. If 𝐽𝑛 (𝑥) denotes Bessel function of order n, prove the followings:


a. cos(𝑥 sin 𝜃) = 𝐽0 (𝑥 ) + 2 ∑∞
𝑛=1 cos(2𝑛𝜃 ) 𝐽2𝑛 (𝑥 )
∞
b. sin(𝑥 sin 𝜃 ) = 2 ∑𝑛=1 sin((2𝑛 − 1)𝜃) 𝐽2𝑛−1 (𝑥 )
𝑑
c. (𝑥 𝜇 𝐽𝜇 (𝑥)) = 𝑥 𝜇 𝐽𝜇−1 (𝑥)
𝑑𝑥
𝑑
d. (𝑥 −𝜇 𝐽𝜇 (𝑥)) = −𝑥 −𝜇 𝐽𝜇+1 (𝑥)
𝑑𝑥
e. 𝐽𝜇′ (𝑥 ) = 2[𝐽𝜇−1 (𝑥 ) − 𝐽𝜇+1 (𝑥)]
𝑥
f. 𝐽𝜇 (𝑥 ) = [𝐽𝜇−1 (𝑥 ) + 𝐽𝜇+1 (𝑥)]
2𝜇

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