Question

The Bessel equation of order n:

0= f(,u — ) + fix + R-1

Please use the Forbenius method (below for when $r_1-r_2=N$ is a positive intiger) to find two solutions as a series in x, when n=1/2.

$y_1=\sum_{n=0}^{\infty}a_nx^{n+r_1}, y_2=Ay_1ln(x)+\sum_{n=0}^{\infty}b_nx^{n+r_2}$

Lastly find these solutions in CLOSED FORM expressions. Please show all steps. DO NOT use Bessel functions .

Answer 1

"We have Bessel differential equation is x^2y"" + xy' + (x^2 - y^2) y = 0 Let y = Sigma^infinity_r = 0 a_r x^m + r y' = Sigma_1 (m + r) a_r x^m = r -1 y"" = Sigma_1 (m + r) a_r x^m + r -1 y"" = Sigma_1(m + r) (m + r - 1) a_r x^m + r -2 sigma_1 (m + r) (m + r - 1) a_r x^m + r - 2 substituting in the given equation Sigma_1 (m + r) (m + r - 1)a_r x^m + n + Sigma_1 (m + r)a_r x^m + r = 0 comparing x^m coefficients m(m - 1)a_0 + m a_0 - n^2 a_0 = 0 (m^2 - n^2) a_0 = therefore a_0 notequalto 0 therefore m^2 = n^2 Rightarrow m = plusminus n comparing x^m + 1 coefficient (m + 1) m a_1 + (m + 1)a_1 - n^2a_1 = 0 (m + 1)^2 - n^2) a_1 = 0 Rightarrow a_1 = 0 comparing x^m + r coefficient "