Internal problem ID [5276]
Book: An introduction to Ordinary Differential Equations. Earl A. Coddington. Dover. NY
1961
Section: Chapter 3. Linear equations with variable coefficients. Page 130
Problem number: 8.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
\[ \boxed {y^{\prime \prime }-2 y^{\prime } x +2 \alpha y=0} \]
✓ Solution by Maple
Time used: 0.062 (sec). Leaf size: 31
dsolve(diff(y(x),x$2)-2*x*diff(y(x),x)+2*alpha*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = c_{1} x \operatorname {KummerM}\left (\frac {1}{2}-\frac {\alpha }{2}, \frac {3}{2}, x^{2}\right )+c_{2} x \operatorname {KummerU}\left (\frac {1}{2}-\frac {\alpha }{2}, \frac {3}{2}, x^{2}\right ) \]
✓ Solution by Mathematica
Time used: 0.014 (sec). Leaf size: 45
DSolve[(1-x^2)*y''[x]-x*y'[x]+\[Alpha]^2*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to c_1 \cosh \left (\alpha \text {arctanh}\left (\frac {x}{\sqrt {x^2-1}}\right )\right )+i c_2 \sinh \left (\alpha \text {arctanh}\left (\frac {x}{\sqrt {x^2-1}}\right )\right ) \\ \end{align*}