# 1. Show that J1;2(x) = (: x ) 1 1 2 sinx and that L1;2(x) = (...

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1. Show that J1;2(x) = (: x ) 1 1 2 sinx and that L1;2(x) = (,,r2 x ) 1 1 2 cosx ( 2 ) 1 1 2 { sin x } Hence, show that J3;2 ( x) = 1rx x- cos x 2. Show that the Bessel equation of order one-half, x 2 y" + xy' + ( x 2 - ) y = 0 can be reduced to the equation V 11 + V = 0 by the change of dependent variable y = x-1 1 2 v(x). From this conclude that Y1(x) = x-1 / 2 cosx and Y2(x) = x-1 ! 2 sinx are solutions of the Bessel equation of order one-half. 3. Show that Jo'(x) = -J1(x) 4. Prove the following results. (a) (c) 1 £ {Jo(x)} = yp2 + 1 and that (b) (d) 5. Evaluate the integral fo,i; Jo(x - t)Jo(t) dt. 1 Ji'(x) = Jo(x) - _!_J1(x) X 1 £ {Io(x)} = yp2 -1 6. (a) Show that Jo (x) = - cos(xsin0) d0. 7T" 0 11,r 211 cosxt (b) Hence, show that J0 (x) = - vT=t2" dt 7T" 0 1 - t2 7. Use the generating function to show that Jn (X + y) = I: Jr(x)Jn-r(Y) r=-oo dF(p) 8. (a) Prove the result .C {xf(x)} = - (b) Use the result in (a) to transform the differential equation d 2 y dy x dx2 + dx + xy = O into another of the first order, and solve it. Then invert to show that the solution of the given differential equation when y(O ) = 1 is Jo (x). 9. Show that the solution of the differential equation is y = AJn(kx) + BYn (kx) 10. Consider the differential equation x 2 y" + cxy' + (x2 - n2 ) y = 0 Let 1- 2r = c. Make the substitution y = x rY and obtain the Bessel differential equation. Hence solve the differential equations (a) x 2 y" + 2xy' + 4x 2 y = 0 2 (b) xy" - 3y' + xy = 0 11. Solve the differential equation x 2 y" - 2xy' + 2 ( x2 + 1) y = 0 12. Solve the following differential equations. (a) xy" - y' + 4x3y = 0 (try x 2 = X) (b) x 2 y" + (2x 2 + x) y' + (2x 2 + x) y = 0 (c) y" + x 2 y = 0 (try x 2 = X) ( d) xy" + ( 1 + 4x2 ) y' + x ( 5 + 4x2 ) y = 0 13. ( a) By a suitable change of variables it is often possible to transform a differential equation with variable coefficients into a Bessel equation of a certain order. For example, show that a solution of x 2 y" + ( a2/32x 2(3 + i- v 2/32 ) y = 0 x > 0 is given by y = x 1l2 f ( ax f3 ) where f (0 is a solution of the Bessel equation of order V. (b) Using the result of Question 14( a), show that the general solution of the Airy equation is y" + xy = 0 X > 0 y = x 112 [cif1 (x 312 ) + c2f2 (x 312)] where fi(O and h() are linearly independent solutions of the Bessel equation of order one-third. 3 14. (a) Show that a solution of x>O is given by y = x 0 f (f3x7), where f (c;) is a solution of the Bessel equation of order n. (b) Using the result of Question 14(a), solve the following differential equations. (i) x 2y" + xy' + 4 (x4 - n 2 ) y = 0 (ii) xy" + y = 0 (iii) x 1f2y 11 + y = 0 15. It can be shown that Jo (x) has infinitely many zeros for x > 0. In particular the first three zeros are approximately 2.405, 5.520, and 8.653. Letting Aj ,j = 1,2, ... denote the zeros of Jo (x), it follows that 1, x= 0, 0 X = l. Verify that y = Jo (AjX) satisfies the differential equation Hence show that I/ 1 / \ 2 y + -y + /\j y = ' 0 X X > 0. if This important property of Jo (>,.j x), known as the orthogonality property, is useful in solving boundary value problems. 4 Hint: Write down the differential equation for J0 ( AiX). Multiply it by xJo (.Xj x) and subtract it from xJo (.Xix) times the differential equation for Jo (.Xj x) Then integrate from O to 1.

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