Internal problem ID [8894]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1315.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, _with_symmetry_[0,F(x)]]]
Solve \begin {gather*} \boxed {x \left (x^{2}-1\right ) y^{\prime \prime }+y^{\prime }+y a \,x^{3}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 45
dsolve(x*(x^2-1)*diff(diff(y(x),x),x)+diff(y(x),x)+y(x)*a*x^3=0,y(x), singsol=all)
\[ y \relax (x ) = c_{1} \sin \left (\frac {\left (x -1\right ) \left (x +1\right ) \sqrt {a}}{\sqrt {x^{2}-1}}\right )+c_{2} \cos \left (\frac {\left (x -1\right ) \left (x +1\right ) \sqrt {a}}{\sqrt {x^{2}-1}}\right ) \]
✓ Solution by Mathematica
Time used: 0.013 (sec). Leaf size: 44
DSolve[a*x^3*y[x] + y'[x] + x*(-1 + x^2)*y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to c_1 \cos \left (\sqrt {a} \sqrt {x^2-1}\right )+c_2 \sin \left (\sqrt {a} \sqrt {x^2-1}\right ) \\ \end{align*}