In Problems 1 and 2, answer true or false without referring back to the text.

1. The general solution of x² y″ + xy′ + (x² − 1)y = 0 is y = c₁J₁(x) + c₂J₋₁(x). _______
2. Since x = 0 is an irregular singular point of x³y″ − xy′ + y = 0, the DE possesses no solution that is analytic at x = 0. _______
3. Both power series solutions of y″ + ln(x + 1)y′ + y = 0 centered at the ordinary point x = 0 are guaranteed to converge for all x in which one of the following intervals?

   (a) (– ∞, ∞)

   (b) (–1, ∞)

   (c) [– , ]

   (d) [–1, 1]

4. x = 0 is an ordinary point of a certain linear differential equation. After the assumed solution y = c_nxⁿ is substituted into the DE, the following algebraic system is obtained by equating the coefficients of x⁰, x¹, x², and x³ to zero:

   

   Bearing in mind that c₀ and c₁ are arbitrary, write down the first five terms of two power series solutions of the differential equation.

5. Suppose the powers series c_k(x − 4)^k is known to converge at −2 and diverge at 13. Discuss whether the series converges at −7, 0, 7, 10, and 11. Possible answers are does, does not, or might.
6. Use the Maclaurin series for sin x and cos x along with long division to find the first three nonzero terms of a power series in x for the function

In Problems 7 and 8, construct a linear second-order differential equation that has the given properties.

7. A regular singular point at x = 1 and an irregular singular point at x = 0.
8. Regular singular points at x = 1 and at x = −3.

In Problems 9–14, use an appropriate infinite series method about x = 0 to find two solutions of the given differential equation.

9. 2xy″ + y′ + y = 0
10. y″ − xy′ − y = 0
11. (x − 1)y″ + 3y = 0
12. y″ − x² y′ + xy = 0
13. xy″ − (x + 2)y′ + 2y = 0
14. (cos x)y″ + y = 0

In Problems 15 and 16, solve the given initial-value problem.

15. y″ + xy′ + 2y = 0,     y(0) = 3,     y′(0) = −2
16. (x + 2)y″ + 3y = 0,     y(0) = 0,     y′(0) = 1
17. Without actually solving the differential equation

    (1 − 2 sin x)y″ + xy = 0

    find a lower bound for the radius of convergence of power series solutions about the ordinary point x = 0.

18. Even though x = 0 is an ordinary point of the differential equation, explain why it is not a good idea to try to find a solution of the initial-value problem

    y″ + xy′ + y = 0,     y(1) = −6,     y′(1) = 3

    of the form c_nxⁿ. Using power series, find a better way to solve the problem.

In Problems 19 and 20, investigate whether x = 0 is an ordinary point, singular point, or irregular singular point of the given differential equation. [Hint: Recall the Maclaurin series for cos x and e^x.]

19. xy″ + (1 − cos x)y′ + x² y = 0
20. (e^x − 1 − x)y″ + xy = 0
21. Note that x = 0 is an ordinary point of the differential equation

    y″ + x²y′ + 2xy = 5 − 2x + 10x³.

    Use the assumption y = c_nxⁿ to find the general solution y = y_c + y_p that consists of three power series centered at x = 0.

22. The first-order differential equation dy/dx = x² + y² cannot be solved in terms of elementary functions. However, a solution can be expressed in terms of Bessel functions.

    (a) Show that the substitution y = − leads to the equation u″ + x²u = 0.

    (b) Use (20) in Section 5.3 to find the general solution of u″ + x²u = 0.

    (c) Use (22) and (23) in Section 5.3 in the forms

    

    and     

    as an aid in showing that a one-parameter family of solutions of dy/dx = x² + y² is given by

    

23. Express the general solution of the given differential equation on the interval (0, ∞) in terms of Bessel functions.
    1. 4x²y″ + 4xy′ + (64x² − 9)y = 0
    2. x²y″ + xy′ − (36x² + 9)y = 0
24. 1. From (34) and (35) of Section 5.3 we know that when n = 0, Legendre’s differential equation (1 − x²)y″ − 2xy′ = 0 has the polynomial solution y = P₀(x) = 1. Use (5) of Section 3.2 to show that a second Legendre function satisfying the DE on the interval (–1, 1) is

       

    2. We also know from (34) and (35) of Section 5.3 that when n = 1, Legendre’s differential equation (1 − x²)y″ − 2xy′ + 2y = 0 possesses the polynomial solution y = P₁(x) = x. Use (5) of Section 3.2 to show that a second Legendre function satisfying the DE on the interval (–1, 1) is

       

    3. Use a graphing utility to graph the logarithmic Legendre functions given in parts (a) and (b).
25. Use binomial series to formally show that

    

26. Use the result obtained in Problem 25 to show that P_n(1) = 1 and P_n(−1) = (−1)ⁿ. See Properties (ii) and (iii) on page 299.
27. The differential equation

    

    is known as Hermite’s equation of order α after the French mathematician Charles Hermite (1822–1901). Show that the general solution of the equation is where

    

    are power series solutions centered at the ordinary point 0.

28. 1. When is a nonnegative integer Hermite’s differential equation always possesses a polynomial solution of degree n. Use given in Problem 27 to find polynomial solutions for and Then use in Problem 27 to find polynomial solutions for and
    2. A Hermite polynomial is defined to be an nth degree polynomial solution of Hermite’s differential equation multiplied by an appropriate constant so that the coefficient of in is Use the polynomial solutions found in part (a) to show that the first six Hermite polynomials are

       

29. Rodrigues’ formula for the Hermite polynomials is

    

    Use this formula to find the six Hermite polynomials listed in part (b) of Problem 28.

30. The differential equation

    

    is known as Laguerre’s equation after the French mathematician Edmond Laguerre (1834–1886). When n is a nonnegative integer, the DE is known to possess polynomial solutions. These solutions are naturally called Laguerre polynomials and are denoted by Rodrigues’ formula for the Laguerre polynomials is

    

    Use this formula to find the Laguerre polynomials corresponding to

31. The differential equation

    

    where α is a parameter, is known as Chebyshev’s equation after the Russian mathematician Pafnuty Chebyshev (1821–1894). Find the general solution of the equation, where and are power series solutions centered at the ordinary point 0 and containing only even powers of x and odd powers of x, respectively.

32. 1. When is a nonnegative integer Chebyshev’s differential equation always possesses a polynomial solution of degree n. Use found in Problem 31 to find polynomial solutions for and Then use in Problem 31 to find polynomial solutions for and
    2. A Chebyshev polynomial is defined to be an nth degree polynomial solution of Chebyshev’s equation multiplied by the constant when n is even and by when n is odd. Use the solutions found in part (a) to obtain the first six Chebyshev polynomials