Question

prove that J2(x)=sum from k=0 to infinity [ (-1)^k/2^9@k+2)*k!(k+2)! ]*x^(2k+2) is a solution of the Bessel differential equation of order 2:

x^2y'' + xy' + (x^2-4)y=0

(-1)4 9- Using the ratio test, one can easily show that the series +2converges for all e R. Prove that (-1)X h(x) = E, 22k +2k!(k + 2)! 22+2 is a solution of the Bessel differential equation of order 2: In(x) is called the Bessel function of the first Remark. In general the function Jn(r) is defined by the power series kind of order n. It can be shown that n(x) is a solution of the Bessel differential equation 22k+nk!(k+n)! y+a+-)y-o. Bessel functions are closely associated with problems possessing circular or cylindrical symmetry, such as the study of free vibrations ofa circular membrane and finding the temperature distribution in a circular cylinder. They are considered to be among the most important unctions in mathematical physics beyond the elementary ones studied in calculus

Answer 1

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