Problem set 6

A review of important results

Ignore this section for now. It is just for my benefit as I realized that you were not taught the three Frobenius theorems in full.


3. Singular point at infinity

Consider the differential equation \begin{equation} \label{q3} x^3 y'' + y = 0. \end{equation}

  1. Use the transformation $x = 1/t$ to show the ODE has a regular singular point at $x = \infty$ and determine its indicial exponents.
  2. Show that one solution to the ODE has a Taylor series at $x = \infty$. Find the series and assuming $y(\infty) = 1$, find $y(1)$ correct to three decimal places.
  3. The leading behaviour of a particular solution of \eqref{q3} is $y \sim x$ as $x \to \infty$. What is the next largest term in the expansion?