TESTETESTEINDEXAÇÃ0 1726751313054 Lista de exercÃ_cios 4 pdf

Lista de exercicios 4 — Pontos singulares (do livro de Boyce e Di Prima) Problems In each of Problems | through 6: 1. 2xy”+y’+xy =0 a. Show that the given differential equation has a regular l singular point at x = 0. 2. xy" +xy'+ (e — 5) =0 b. Determine the indicial equation, the recurrence relation, and , the roots of the indicial equation. 3. xy”+y=0 c. Find the series solution (x > 0) corresponding to the larger 4. xy”+y'-y=0 root. 5. x°y" +xy' +(x —2)y =0 d. If the roots are unequal and do not differ by an integer, find 6 ” I , =0 the series solution corresponding to the smaller root also. say t+ (Lay y= 7. The Legendre equation of order a is a. Show that x = 0 is a regular singular point. b. Show that the roots of the indicial equation are r; = r2 = 0. 2)" ! _ 1 2 (l-x)y" —2xy tala + Dy = 0. c. Show that one solution for x > 0 is The solution of this equation near the ordinary point x = 0 was oo (=1)"x2" discussed in Problems 17 and 18 of Section 5.3. In Example 4 of Jo(x) = 1+ S> mann” Section 5.4, it was shown that x = +1 are regular singular points. n=l 2i(nl) a. Determine the indicial equation and its roots for the point . . . . yell. The function Jp is known as the Bessel function of the first kind b. Find a series solution in powers of x — 1 forx —1> 0. “ one “he h ies f for all Hint: Write 1 + x = 2 +(x — 1) andx =14+(x— 1). . ow that the series for Jo(x) converges for all x. Alternatively, make the change of variable x — 1 = f and 11. Referring to Problem 10, use the method of reduction of order determine a series solution in powers of f. to show that the second solution of the Bessel equation of order zero 8. The Chebyshev equation is contains a logarithmic term. Hint: If yo(x) = Jo(x) v(x), then (1—x*)y"—xy' +a°y =0, dx y2(2) = Jo(x) / ———,. where a is a constant; see Problem 8 of Section 5.3. x (J (x )) a. Show that x = 1 and x = —1 are regular singular points, and 1 find the exponents at each of these singularities. Find the first term in the series expansion of ———,. b. Find two solutions about x = 1. x (Jo(x)) 9. The Laguerre!? differential equation is 12. The Bessel equation of order one is Qo” , 2 _ xy” +(1—x)y'+Ay =0. xy’ +xy +(x -Dy=0. . . . a. Show that x = 0 is a regular singular point. a. Show that x = 0 is a regular singular point. b. Show that the roots of the indicial equation are r, = 1 and b. Determine the indicial equation, its roots, and the recurrence m=. relation. c. Show that one solution for x > 0 is c. Find one solution (for x > 0). Show that if A =m, a oo positive integer, this solution reduces to a polynomial. When x (—1)"x2" : . ae J\(x) == S> ——_———_.. properly normalized, this polynomial is known as the Laguerre 2 (n+ 1)!n!22" polynomial, L,,(x). n=0 10. The Bessel equation of order zero is The function J; is known as the Bessel function of the first kind 5 5 of order one. xy" + xy +x°y =0. d. Show that the series for J;(x) converges for all x. e. Show that it is impossible to determine a second solution of the form 13Edmond Nicolas Laguerre (1834-1886), a French geometer and analyst, 1 oe studied the polynomials named for him about 1879. He is also known for an x S> bnx", x > 0. algorithm for calculating roots of polynomial equations. n=0