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Question

a) Consider the differential equation x(x-1) y^{\prime \prime}+\left(\frac{7}{2} x-\frac{3}{2}\right) y^{r}+\frac{3}{2} y=0 Specify what type of differential equation it is, and show that it has a regular singular point at x = 0. (b) Show that the indicial equation has two solutions a = 0 and a -12 (c) A solution of the equat ion corres pondingto a = -has the form y(x)=x^{-\frac{1}{2}} \sum_{n=0}^{\infty} a_{n} x^{n} (d) Deduce that y(x)=\frac{a_{0}}{x^{\frac{1}{2}}(1-x)} where ao is an arbitrary constant. \text { where } a_{0} \neq 0 . \text { Show that } a_{n+1}=a_{n} \text { where } n \geq 0