Question

Use Fuchs' theorem to solve the following ODE: In addition to obtaining both solutions by application of Fuchs' theorem, show that the 'second' solution can also be obtained from the first using the Wronskian. That is, two linearly independent solutions gi(x) and y2(x) to the ODE are (nonlinearly) related by I(2)dr dx2 , where Recall that the additive constants associated with the two integrals can be omit ted

Answer 1

Given that ODE - xy^'' + y^' = 0 - (i) suppose y = sigma^infinity_n = 0 a_n x^n + r is solution of ODE equation . y = sigma^infinity_n = 0 a_n x^n + r - (ii) y^' = sigma^infinity_n = 0 a_n (n + r) x^n + r - 1 - (iii) y^'' = sigma^infinity_n = 0 a_n (n + r) (n + r - 1) x^n + r - 2 - (iv) equ^n (i) and equ^n (iii) & equ^n (iv) x sigma^infinity_n = 0 a_n (n + r) (n + r -1) x^n + r - 2 + sigma^infinity_n = 0 a_n x^n + r - 1 (n + r) = 0 sigma