Question

need answers to both!

16. [-/1 Points] DETAILS ZILLDIFFEQMODAP116.4.002. Use (1) in Section 6.4. x2y" + xy + (r? - v2y = 0 (1) Find the general solution of the given differential equation on (0,co). (The definitions of various Bessel functions are given here.) x@y" + xy' + (x2 - 4)y - 0 O C22(x) + C/(x) OC}(x) + C,Y-2(x) OC+2(X) + C22C%) O +2(x) + C22-2(x) OCP-() + C7,6%) 17. (-/1 Points] DETAILS ZILLDIFFEQMODAP11 6.4.004. Use (1) in Section 6.4. x?y" + xy' + (x2 - v?y - 0 (1) Find the general solution of the given differential equation on (0,0). (The definitions of various Bessel functions are given here.) 64x2y" + 64xy' + (64x2 - 1)y = 0 O y(x) - C,11/8(x) + CK 1/8(*) O y(x) - (x) + Cy?-,(*) O x(x) - C+1/64(x) + C, 1/646X) O y(x) - C'1/(x) + C2-3/06) O y(x) - 1/646X) + C2-1/64(*)

Answer 1

Now depending on ve the are two cases 10 Bessel's differential equation is given by x*y" + xy + (x2- Dy zo soladeben to Bersel's differential equation has different solution. The number u is called the order of Bessel's differential equation. There i) The order it is non- integer (i) The order se is integer The order ie is non-integer The general solution of Berse's equation when u is a • integes and positive is given by g(x) = C. J (x) + C₂ I (s). Where C, and are constants. JJ CK) and Jv (x) an Case 1 non - are Bessel's function of first kind. 21 Let's see what the expression of Ju cx) and Javcx). 2ptv Jucx) = pro ipt [p+v+1 where Ipti is nothing but gamama function of pri and (pri=p! 2p-v p=o ipti Ip-vel 38 J (2) P (1-) کے 2 -V

Camlin Page Date Note that when & is non-integer Jj (x) and indepent functions. But when e is a integel Jv (x) and J., (se) depends second J (W) other. on case each arises ii) when the order a is an integer. The general soln e is an of Bessel's D. E. When ent given by g(x) + C, Yv (oc). where Yv(x) is Bessel's function of second kind. Yv(x) be expressed function of Jucas and Javca)! égei .C J, (x) can Yv(x) Jv (2) COSTTV - Joulx) Sin TV. Yv (2) as above makes Jy (x) and By defining Yuck) widependent function examine Now lets ou given Bessel's D.E i) x²y + xy + (x²-4)=0. [w? 4 => U=2 This is a Bessel's equation of order 2; an integer, so the solution general this Bessel's D-G for is y (x) C Jz (x) + C2 Y₂(*) which is option no: 2.3

out we get ii) 64 x² " +64 x 4' + (64x² - D j = 0 Dividing by 64 through x²g" + xy' + (x² - 4) = 0 xiya + xy + (22-11-)") = 0 of ordes which Heu Bessel's equation is non integer D.E in So the general solution to the Bessel's given by . ya) = C, Jy (on) + C₂ J (x) option no: 4 which