TESTETESTEINDEXAÇÃ0 1726751312393 Lista de exercÃ_cios 2 pdf

Lista de exercicios 2 (do livro de Boyce e Di Prima) In each of Problems 1 through 11: 17. Show directly, using the ratio test, that the two series solutions a. Seek power series solutions of the given differential equation of Airy’s equation about x = 0 converge for all x; see equation (20) about the given point xg; find the recurrence relation that the of the text. coefficients must satisfy. . . 18. The Hermite Equation. The equation b. Find the first four nonzero terms in each of two solutions y; ” , and y» (unless the series terminates sooner). yo —2xy +Ay=0, —0o <x < 00, c. By evaluating the Wronskian W[y,, y2](xo), show that y, where \ is a constant, is known as the Hermite? equation. It is an and y> form a fundamental set of solutions. important equation in mathematical physics. d. If possible, find the general term in each solution. a. Find the first four nonzero terms in each of two solutions 1. y’-y=0, x =0 about x = 0 and show that they form a fundamental set of 2. y’+3y'=0, x =0 solutions. 3. y”—xy'-y=0, x=0 b. Observe that if A is a nonnegative even integer, then one 4 "_yyloya 0. x =l or the other of the series solutions terminates and becomes a 7 y > 7 m0 polynomial. Find the polynomial solutions for A = 0, 2, 4, 6, 5. y"+kx?y=0, x9 =0, kaconstant ae . . 1 > *0 , 8, and 10. Note that each polynomial is determined only up to a 6. ql —* yy ty=0, x%=0 multiplicative constant. 7. y"+xy'+2y=0, x =0 c. The Hermite polynomial H,,(x) is defined as the polynomial 8. xy" +y'+txy=0, x =1 solution of the Hermite equation with 4 = 2n for which the 9. (3—x)y"—3xy'-y=0, x=0 coefficient of x” is 2”. Find Ho(x), Hi(x), ... , H5(x). 10. 2y”’+xy'+3y=0, x) =0 19. Consider the initial-value problem y’ = \/1 — y?, y(0) = 0. ll. 2y"4(x+4+)Dy'4+3y=0, x =2 a. Show that y = sinx is the solution of this initial-value blem. In each of Problems 12 through 14: Pro ean ns : . : b. Look for a solution of the initial-value problem in the form of a. Find the first five nonzero terms in the solution of the given . . . initial-value problem. a power series about x = 0. Find the coefficients up to the term @ b. Plot the four-term and the five-term approximations to the in x" in this series. solution on the same axes. In each of Problems 20 through 23, plot several partial sums in a c. From the plot in part b, estimate the interval in which the series solution of the given initial-value problem about x = 0, four-term approximation is reasonably accurate. thereby obtaining graphs analogous to those in Figures 5.2.1 through 12. y"—xy'—y=0, (0) =2, y'(0) =1: see Problem 3 5.2.4 (except that we do not know an explicit formula for the actual . a 7 — solution). 13. y” '+2y =0, =4, y(0) =-1; Problem 7 S yorry * y=0, (0) y(0) , see Bropiem © 20. y”+xy' +2y =0, y(0) =0, y'(0) = 1; see Problem 7 » (l- ’ ‘-y=0, 0) = -3, 0) =2 i” ( baa, . . . x - * , Seccumme OF 2 4a)" +27 =0, 1) =, YO =1 . a. By making the change of variable x — 1 = t and assuming nd _ VAY 1. that y has a Taylor series in powers of f, find two series solutions @ 22. y"’+x'y=0, (0) = 1, y'(0) =0; see Problem 5 of © 23. (1-x)y”’+xy’-2y =0, y(0) =0, y(0) =1 y+ (x= Diy +08 Dy =0 snunsnnnnnninnnnnnnsnnnsnnnnnnsnnnnnnsnnsnnnnsnn in powers of x — 1. >Charles Hermite (1822-1901) was an influential French analyst and b. Show that you obtain the same result by assuming that y algebraist. An inspiring teacher, he was professor at the Ecole Polytechnique has a Taylor series in powers of x — 1 and also expressing the and the Sorbonne. He introduced the Hermite functions in 1864 and showed in coefficient x2 — 1 in powers of x — 1 1873 that e is a transcendental number (that is, e is not a root of any polynomial : P , equation with rational coefficients). His name is also associated with Hermitian 16. Prove equation (10). matrices (see Section 7.3), some of whose properties he discovered.