 Solution

ID: past-exam-of-the-options-ii-course-of-the-university-of-oxford/2012/j/j2/a/ii/solution

Differentiate the power series twice and align the coefficient of

ξ^{k}

. The equation becomes

(k + 2) (k + 1) a_{k + 2} + (ϵ - 1 - 2 k) a_{k} = 0 (k \geq 0) .

Thus

a_{k + 2} = \frac{2 k + 1 - ϵ}{( k + 2 ) ( k + 1 )} a_{k} .

Even coefficients are generated from

a_{0}

, and odd coefficients from

a_{1}

 New to topics? Read the docs here!