Lista 3 - Equação de Euler - Cálculo 4

Lista de exercicios 3 — Equacdo de Euler (do livro de Boyce e Di Prima) Problems In each of Problems | through 8, determine the general solution of the In each of Problems 9 through 11, find the solution of the given initial- given differential equation that is valid in any interval not including value problem. Plot the graph of the solution and describe how the the singular point. solution behaves as x — 0. 1. xy" +4xy’ +2y =0 @ 9. 2x?y"4+xy'-3y=0, y=1, yl) =4 2. (x +1)?y" +3(x + ly’ +0.75y = 0 © 10. 4x°y"+8xy’+17y=0, y() =2, y() =-3 3. xy” —3xy’ +4y =0 @ ll. x’y"—3xy'+4y=0, y(-1) =2, y(-1) =3 4, x°y”—xy'+y=0 In each of Problems 12 through 23, find all singular points of the given 5. x?y"+ 6xy’—y =0 equation and determine whether each one is regular or irregular. 6. 2x?y" — 4xy’ + 6y = 0 12. xy”+(1—x)y’+xy =0 7. xy ~ Say’ 9y = 0 / 13. x2(1 — x)*y" + 2xy’ +4y =0 8. (x —2)*y" +5(x — 2) y+ 8y =0 2 _ ” _ yo _ Co 14, x81 —x)y" + (x ~ 2y! — 3xy = 0 form }~ a,x". Show that (except for constant multiples) there is only 2 =0 2 2) ,,/ / _— n IS. x81 —x*)y" + (2) yit4y=0 one nonzero solution of this form in Problem 30 and that there are ayo oy _ no nonzero solutions of this form in Problem 31. Thus in neither case 16. C vy + x vy +U+xHy=0 ; can the general solution be found in this manner. This is typical of 17. x*y”+xy' +(x* —v*)y =0 (Bessel equation) equations with singular points. 18. (x +2)7(x — ly” + 3(« — Dy’ — (x +-2)y =0 30. 2xy" +3y’+xy =0 19. x(3—x)y"+(x+)y’-2y =0 31. 2x?y"4+3xy’-(1+x)y =0 20. xy” +e*y’ + (3cosx)y =0 32. Singularities at Infinity. The definitions of an ordinary point 21.) y”+(In|x|) yy’ + 3xy =0 and a regular singular point given in the preceding sections apply 22. (sinx)y"” +xy' +4y =0 only if the point xo is finite. In more advanced work in differential 3B . 4 3y! =0 equations, it is often necessary to consider the point at infinity. This > (xsinx)y" +3y +xy = is done by making the change of variable € = 1/x and studying the 24, Find all values of a for which all solutions of resulting equation at € = 0. Show that, for the differential equation xy" + axy! + = 0 approach zero as x > 0. P(x)y” + OC) y' + R(x) y = 0, 25. Find all values of 3 for which all solutions of the point at infinity is an ordinary point if x?y” + By = 0 approach zero as x > 0. . . ae 1 2P(1/g) — Q(1/£) RU/8) 26. Find ¥ so that the solution of the initial-value problem PaUaA\ ee and APD x?y"” —2y =0, y(1) = 1, y'(1) =7 is bounded as x > 0. (1/8) g g g*PC1/§) 27. Consider the Euler equation x7y” + axy’ + Gy = 0. Find have Taylor series expansions about € = 0. Show also that the point at conditions on a and 3 so that: infinity is a regular singular point if at least one of the above functions a. All solutions approach zero as x > 0. does not have a Taylor series expansion, but both b. All solutions are bounded as x > 0. é 2P(1/é) Q(1/) R(A/E) c. All solutions approach zero as x — oo. DTA Geo - ou) and = PC 2 2P(1 d. All solutions are bounded as x — ov. C/6) $ é g°PC1/6) e. All solutions are bounded both as x — 0 and as x > w. do have such expansions. 28. Using the method of reduction of order, show that if r; is a —_In each of Problems 33 through 37, use the results of Problem 32 to repeated root of determine whether the point at infinity is an ordinary point, a regular ingul int. i lar singul int of the given differential rr—l) tar¢6=0, singular point, or an irregular singular point of the given differenti equation. then x”! and x"! In x are solutions of x?y"+axy’+Gy =Oforx>0. 33, y"+y=0 29. Verify that W[x* cos(j Inx), x* sin(ye Inx)] = ppx?—!, 34. x?y” + xy’ —4y =0 In each of Problems 30 and 31, show that the point x= Oisa 35. (1—x?)y” —2xy’ Hala tly =0 (Legendre equation) regular singular point. In each problem try to find solutions of the 36. y’—2xy’'+Ay=0 (Hermite equation) 37. y’—xy =0_ (Airy equation)