Internal problem ID [8951]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1372.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {y^{\prime \prime }+\frac {2 x y^{\prime }}{x^{2}-1}+\frac {\left (\left (x^{2}-1\right ) \left (a \,x^{2}+b x +c \right )-k^{2}\right ) y}{\left (x^{2}-1\right )^{2}}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.102 (sec). Leaf size: 117
dsolve(diff(diff(y(x),x),x) = -2*x/(x^2-1)*diff(y(x),x)-((x^2-1)*(a*x^2+b*x+c)-k^2)/(x^2-1)^2*y(x),y(x), singsol=all)
\[ y \relax (x ) = c_{1} {\mathrm e}^{\sqrt {-a}\, x} \HeunC \left (4 \sqrt {-a}, k , k , 2 b , \frac {k^{2}}{2}+a -b +c , \frac {x}{2}+\frac {1}{2}\right ) \left (x^{2}-1\right )^{\frac {k}{2}}+c_{2} {\mathrm e}^{\sqrt {-a}\, x} \HeunC \left (4 \sqrt {-a}, -k , k , 2 b , \frac {k^{2}}{2}+a -b +c , \frac {x}{2}+\frac {1}{2}\right ) \sqrt {2 x -2}\, \left (x +1\right )^{-\frac {k}{2}} \left (x -1\right )^{\frac {k}{2}-\frac {1}{2}} \]
✓ Solution by Mathematica
Time used: 0.198 (sec). Leaf size: 189
DSolve[y''[x] == -(((-k^2 + (-1 + x^2)*(c + b*x + a*x^2))*y[x])/(-1 + x^2)^2) - (2*x*y'[x])/(-1 + x^2),y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to e^{\sqrt {-a} x} (x+1)^{-k/2} \left (c_1 (x+1)^{k/2} \left (x^2-1\right )^{k/2} \text {HeunC}\left [(k+1) \left (2 \sqrt {-a}-k\right )-a+b-c,2 \left (2 \sqrt {-a} (k+1)+b\right ),k+1,k+1,4 \sqrt {-a},\frac {x+1}{2}\right ]+\sqrt {2} c_2 (x-1)^{k/2} \text {HeunC}\left [-2 \sqrt {-a} (k-1)-a+b-c,2 \left (2 \sqrt {-a}+b\right ),1-k,k+1,4 \sqrt {-a},\frac {x+1}{2}\right ]\right ) \\ \end{align*}