2. (a) The spherical Bessel's functions jp(x) are solutions of the equation xy + 2xy' +...

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2. (a) The spherical Bessel's functions jp(x) are solutions of the equation x²y" + 2xy' + (x² − p(p+1))y = 0, where p is a positive integer constant or zero. For a series solution of equation (2), derive the indicial equation, and show that it has one root equal to p and one negative root. Show that the root equal to p leads to a solution jp(x) of equation (2) of the form ∞ İp(x) = x²(1 + Σanx"). n=1 (2) Find the general recurrence relation for the coefficients an and show that an = 0 when n is odd. = 1/5!. In the case p = 0, show that a2 = − -1/3! and a4 = (b) Give an example of a second order linear differential equation with polynomial coefficients possessing exactly three singular points such that two points x = ±4 are irregular singular points and the point x = 1 is a regular singular point. Justify your example. 2. (a) The spherical Bessel's functions jp(x) are solutions of the equation x²y" + 2xy' + (x² − p(p+1))y = 0, where p is a positive integer constant or zero. For a series solution of equation (2), derive the indicial equation, and show that it has one root equal to p and one negative root. Show that the root equal to p leads to a solution jp(x) of equation (2) of the form ∞ İp(x) = x²(1 + Σanx"). n=1 (2) Find the general recurrence relation for the coefficients an and show that an = 0 when n is odd. = 1/5!. In the case p = 0, show that a2 = − -1/3! and a4 = (b) Give an example of a second order linear differential equation with polynomial coefficients possessing exactly three singular points such that two points x = ±4 are irregular singular points and the point x = 1 is a regular singular point. Justify your example.