University:
Amherst College
Course:
MATH 260 | Differential Equations
Academic year:

2023
Views:

29

Pages:

2
Author:

Alexa Livingston

6.5. EXERCISES 6.5 Exercises 1. Let the functions f, g : < → < be defined as follows  sin t   , t 6= 0 t f (t) = ,   1, t=0 g(t) = Show that f and g are analytic at t = 0.  1 − cos t  , t 6= 0   t2    1 , 2 t=0 2. Find the series solution of the following IVPs (i) ẍ + tẋ − 2x = 0, x(0) = 1, ẋ(0) = 0 (ii) t(2 − t)ẍ − 6(t − 1)ẋ − 4x = 0, x(1) = 1, ẋ(1) = 0 (iii) ẍ + et ẋ + (1 + t2 )x = 0, x(0) = 1, ẋ(0) = 0 (iv) ẍ − (sin t)x = 0, x(π) = 1, ẋ(π) = 0 3. Show that the general solution of the Chebyshev differential equation (1 − t2 )ẍ − tẋ + a2 x = 0 is given by x(t) = c0 " + c1 " ∞ X (−a)2 (22 − a2 )......((2n − 2)2 − a2 ) 2n 1+ t (2n)! n=1 # ∞ X (1 − a2 )(32 − a2 )......((2n − 1)2 − a2 ) 2n+1 t+ t (2n − 1)! n=1 # 4. Show that the two linearly independent solutions of the following differential equation 1 1 t2 ẍ + t(t − )ẋ + x = 0 2 2 are given by x1 (t) = |t| x2 (t) = |t| ∞ X (−1)n n=0 ∞ X 1 2 n=0 (2t)n (2n + 1)(2n − 1)......3.1 (−1)n tn n! 5. Show that the two linearly independent solutions of the differential equation t(1 − t)ẍ + (1 − t)ẋ − x = 0 are given by x1 (t) x2 (t) = 1+ ∞ X 1.2.5......((n − 1)2 + 1) n t (n!)2 n=1 ∞ X 1.2.5.....((n − 1)2 + 1) = x1 (t)ln|t| + 2 (n!)2 n=1 ! n X k−2 × tn k((k − 1)2 + 1) ! k=1 6. Using Rolle’s theorem show that between two consucative zeros of Jn (t), there exists precisely one zero of Jn+1 (t). 7. Prove the following identities for Bessel functions Jµ (t), µ ∈ <. R (i) tµ Jµ−1 (t)dt = tµ Jµ (t) + c R (ii) t−µ Jµ+1 (t)dt = −t−µ Jµ (t) + c R R (iii) Jµ+1 (t)dt = Jµ−1 (t)dt − 2Jµ (t) 8. Show that 2 kJn (klm t)k = Z 1 0 tJn2 (klm t)dt = 1 2 J (klm ) 2 n+1 9. Using Rodrigues formula, show that  0, n is odd   Pn (0) =   (−1)n/2 1.3....n − 1 , n is even 2.4.....n 10. Let p(t) be a polynomial of degree n. Show that p(t) is orthogonal to all Legendre polynomials of degree strictly less than n.

6.5 Series Solution Exercises

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