Internal problem ID [8828]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1248.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {\left (x^{2}-1\right ) y^{\prime \prime }+a x y^{\prime }+\left (x^{2} b +x c +d \right ) y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.063 (sec). Leaf size: 150
dsolve((x^2-1)*diff(diff(y(x),x),x)+a*x*diff(y(x),x)+(b*x^2+c*x+d)*y(x)=0,y(x), singsol=all)
\[ y \relax (x ) = c_{1} {\mathrm e}^{\sqrt {-b}\, x} \left (x^{2}-1\right )^{-\frac {a}{4}} \left (\left (x -1\right ) \left (x +1\right )\right )^{\frac {a}{4}} \HeunC \left (4 \sqrt {-b}, \frac {a}{2}-1, \frac {a}{2}-1, 2 c , d -c -\frac {a^{2}}{8}+b +\frac {1}{2}, \frac {x}{2}+\frac {1}{2}\right )+c_{2} {\mathrm e}^{\sqrt {-b}\, x} \left (\frac {x}{2}+\frac {1}{2}\right )^{1-\frac {a}{4}} \left (\frac {x}{2}-\frac {1}{2}\right )^{\frac {a}{4}} \left (x^{2}-1\right )^{-\frac {a}{4}} \HeunC \left (4 \sqrt {-b}, 1-\frac {a}{2}, \frac {a}{2}-1, 2 c , d -c -\frac {a^{2}}{8}+b +\frac {1}{2}, \frac {x}{2}+\frac {1}{2}\right ) \]
✓ Solution by Mathematica
Time used: 0.173 (sec). Leaf size: 192
DSolve[(d + c*x + b*x^2)*y[x] + a*x*y'[x] + (-1 + x^2)*y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {1}{2} e^{\sqrt {-b} x} \left (c_2 (x-1)^{a/4} \left (x^2-1\right )^{-a/4} (x+1)^{1-\frac {a}{4}} \text {HeunC}\left [\frac {1}{4} a \left (a-4 \sqrt {-b}-2\right )-b+4 \sqrt {-b}+c-d,2 \left (2 \sqrt {-b}+c\right ),2-\frac {a}{2},\frac {a}{2},4 \sqrt {-b},\frac {x+1}{2}\right ]+2 c_1 \text {HeunC}\left [a \sqrt {-b}-b+c-d,2 \left (a \sqrt {-b}+c\right ),\frac {a}{2},\frac {a}{2},4 \sqrt {-b},\frac {x+1}{2}\right ]\right ) \\ \end{align*}