QuestionQuestion

Transcribed TextTranscribed Text

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.

Solution PreviewSolution Preview

This material may consist of step-by-step explanations on how to solve a problem or examples of proper writing, including the use of citations, references, bibliographies, and formatting. This material is made available for the sole purpose of studying and learning - misuse is strictly forbidden.

    $40.00 for this solution

    PayPal, G Pay, ApplePay, Amazon Pay, and all major credit cards accepted.

    Find A Tutor

    View available Differential Equations Tutors

    Get College Homework Help.

    Are you sure you don't want to upload any files?

    Fast tutor response requires as much info as possible.

    Decision:
    Upload a file
    Continue without uploading

    SUBMIT YOUR HOMEWORK
    We couldn't find that subject.
    Please select the best match from the list below.

    We'll send you an email right away. If it's not in your inbox, check your spam folder.

    • 1
    • 2
    • 3
    Live Chats