Use the method of Frobenius to find two linear independent series solution to Bessel’s equation of order p = 0 x 2 y ′′ + xy′ + x 2 y = 0, x > 0.

[10]

(ii) Let Jp(x) denote a Bessel function of the first kind of order p. Prove the following identities: d dx (x pJp(x)) = x pJp−1(x), and d dx ( x −pJp(x) ) = −x −pJp+1(x) The using the above identities to prove that Jp+1(x) = 2p x Jp(x) − Jp−1(x)

Use (ii) to to rewrite the Bessel function of zero order and first kind J0(x) in terms of J−1(x) and J−2(x)