Internal problem ID [10986]
Book: Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev.
Second edition
Section: Chapter 2, Second-Order Differential Equations. section 2.1.2-5 Equation of form
\((a x^2+b x+c) y''+f(x)y'+g(x)y=0\)
Problem number: 152.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [_Gegenbauer, [_2nd_order, _linear, `_with_symmetry_[0,F(x)]`]]
\[ \boxed {\left (-x^{2}+1\right ) y^{\prime \prime }-y^{\prime } x +y n^{2}=0} \]
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 33
dsolve((1-x^2)*diff(y(x),x$2)-x*diff(y(x),x)+n^2*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = c_{1} \left (x +\sqrt {x^{2}-1}\right )^{-n}+c_{2} \left (x +\sqrt {x^{2}-1}\right )^{n} \]
✓ Solution by Mathematica
Time used: 0.139 (sec). Leaf size: 91
DSolve[(1-x^2)*y''[x]-x*y'[x]+n^2*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to c_1 \cosh \left (\frac {1}{2} n \left (\log \left (1-\frac {x}{\sqrt {x^2-1}}\right )-\log \left (\frac {x}{\sqrt {x^2-1}}+1\right )\right )\right )-i c_2 \sinh \left (\frac {1}{2} n \left (\log \left (1-\frac {x}{\sqrt {x^2-1}}\right )-\log \left (\frac {x}{\sqrt {x^2-1}}+1\right )\right )\right ) \]