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.
Fig: 1
Fig: 2
Fig: 3
Fig: 4
Fig: 5
Fig: 6
Fig: 7
Fig: 8
Fig: 9
Fig: 10
Fig: 11
Fig: 12