Internal problem ID [6029]
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: 7.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [_Gegenbauer, [_2nd_order, _linear, `_with_symmetry_[0,F(x)]`]]
\[ \boxed {\left (-x^{2}+1\right ) y^{\prime \prime }-y^{\prime } x +\alpha ^{2} y=0} \]
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 33
dsolve((1-x^2)*diff(y(x),x$2)-x*diff(y(x),x)+alpha^2*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = c_{1} \left (x +\sqrt {x^{2}-1}\right )^{-\alpha }+c_{2} \left (x +\sqrt {x^{2}-1}\right )^{\alpha } \]
✓ Solution by Mathematica
Time used: 0.089 (sec). Leaf size: 91
DSolve[(1-x^2)*y''[x]-x*y'[x]+\[Alpha]^2*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to c_1 \cosh \left (\frac {1}{2} \alpha \left (\log \left (1-\frac {x}{\sqrt {x^2-1}}\right )-\log \left (\frac {x}{\sqrt {x^2-1}}+1\right )\right )\right )-i c_2 \sinh \left (\frac {1}{2} \alpha \left (\log \left (1-\frac {x}{\sqrt {x^2-1}}\right )-\log \left (\frac {x}{\sqrt {x^2-1}}+1\right )\right )\right ) \]