 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-2-of-the-natural-sciences-course-of-the-university-of-cambridge/2020/paper-1/2/a/i/solution

Differentiate the power series term by term inside its radius of convergence, and reindex the second derivative so that every coefficient multiplies

x^{m}

\sum_{m = 0}^{\infty} [(m + 2) (m + 1) a_{m + 2} + {n (n + 1) - m (m + 1)} a_{m}] x^{m} = 0.

a_{m + 2} = \frac{m ( m + 1 ) - n ( n + 1 )}{( m + 1 ) ( m + 2 )} a_{m} = \frac{( m - n ) ( m + n + 1 )}{( m + 1 ) ( m + 2 )} a_{m}, m \geq 0.

The even coefficients depend only on

a_{0}

and the odd coefficients only on

a_{1}

x = 0

 New to topics? Read the docs here!