(i) Consider the differential equation (x+1)^{2} \frac{d^{2} y}{d x^{2}}-4(x+1) \frac{d y}{d x}+6 y=0 (a) Show that y1(x) = (x + 1)² is a solution of this ODE. (b) By using

the reduction of order method find a linearly independent second solution y2(x), and write down the general solution. (ii) Consider the differential equation x \frac{d^{2} y}{d x^{2}}+(1-x) \frac{d y}{d x}+\lambda y=0 where A is a constant. (a) Show that x = 0 is a regular singular point. (b) By writing a series solution in the form y=\sum_{n=0}^{\infty} a_{n} x^{n+\alpha} show that the indicial equation is satisfied by a =0, and find the recurrence relation satisfied by the coefficients a,: (c) Show that if A m, a positive integer, one solution reduces to a polynomial, and find that solution for m = 3.