Question

Show that tₒ =1 is a regular singular point of the ODE (t−1) u¨ +(t² −1)u˙ +u=0, derive the associated Euler equations and solve the ODE with a suitable power series ansatz centered it tₒ =1, for t>1, by deriving a well-defined recursion relation on its coefficients aₙ .

Answer 1

To solve the ODE given with a regular singular point at t_0 = 1, we first prove that it is indeed a regular singular point. Then an associated Euler equation is derived, and the ODE is solved using a power series ansatz centered at t_0 = 1, resulting in a recursion relation for the coefficients a_n.

Step-by-step explanation:

The given ordinary differential equation (ODE) is:

(t-1)u´´ + (t²-1)u´ + u = 0.

To show that t_0 = 1 is a regular singular point, we check the coefficients of the derivatives after dividing by (t-1), the term with the lowest power of t. If these transformed coefficients are analytic at t_0 = 1, then it is a regular singular point. An Euler equation can be derived by replacing t-1 with z, leading to a form that assumes u'' + P(z)u' + Q(z)u = 0, with P(z) and Q(z) analytic at z = 0.

To solve the ODE, we use a power series ansatz centered at t_0 = 1 for t > 1. Assuming a solution u(t) in the form of a power series, u(t) = ∑a_n(t-1)^n, and substituting it into the ODE leads to a recursion formula that relates the coefficients a_n.