solve please

Q.1 (1) Find the radius, center and interval of convergence of the power series 2m=2 (-1)m(x-1)2m 4 m (8) (ii) Derive the Legendre polynomials Ps (x) and P. (x) using Rodrigues's formula and show that these polynomials are the solution of Legendre equation. (8) (iii Find the inverse Laplace Transform of the following functions; (a) F(S)= 15 $2+45+29 using first shift theorem. (8) 1 (b) F(S) =4+s' using Transform of integration. (8) (c) F(S) = S ($3+1)(83+5) . by convolution. (8) [CLO1, Cognitive, Level 1] Q.2 Find a general solution of the following ODEs (a) (xD] + 4D?)y = Ze-X, using Undetermined Coefficients method. (b) y"" + 4y" + 85y' = 135xe*, by Variation of Parameter. [CLO2, Cognitive, Level 1] (20) Q.3 (a) Evaluate the integral fox]/, (x)dx , where /2(x) is the Bessel function of the first kind. [CLOZ, Cognitive, Level 3] (b) Use the given transformation to reduce the following ODE into a Bessel differential equation and find the general solution, if possible xy" - y' +4xy =0, y=ux, 7 = 2x. [CLO2, Cognitive, Level ?] (20)