Reference no: EM133106769

The Bessel function of the first kind, Jo(x), is a solution of the following ODE

(xy' (x))' + xy(x) = 0

satisfying the boundary condition y(0) = 1 and y'(0) = 0.

Question 1: Show that the Laplace transform of J₀ is

L(J₀)(s) = 1/√(1+s²)

Question 2: Solve the Volterra integral equation

₀∫^xJ_o (x - t) y(t) dt = sin x, x ≥ 0.

Question 3: Show that the Laplace transform of J₀ can be rewritten as

∑_n=0^∞α_n/s²ⁿ⁺¹

where

α_n = (-1)ⁿ(2n)!/(4ⁿ(n!)²)

Question 4: By applying the inverse Laplace transform to the series in part (c), derive the following series for J₀

J₀(x) = ∑_n=0^∞(-1)ⁿ/4ⁿ(n!)²