Internal problem ID [8993]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1414.
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 {\left (-a^{2} \left (\sinh ^{2}\relax (x )\right )-n \left (n -1\right )\right ) y}{\sinh \relax (x )^{2}}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.125 (sec). Leaf size: 97
dsolve(diff(diff(y(x),x),x) = -(-a^2*sinh(x)^2-n*(n-1))/sinh(x)^2*y(x),y(x), singsol=all)
\[ y \relax (x ) = c_{1} \left (\sinh ^{n}\relax (x )\right ) \hypergeom \left (\left [-\frac {a}{2}+\frac {n}{2}, \frac {a}{2}+\frac {n}{2}\right ], \left [\frac {1}{2}\right ], \frac {\cosh \left (2 x \right )}{2}+\frac {1}{2}\right )+\frac {c_{2} \left (2 \cosh \left (2 x \right )+2\right )^{\frac {3}{4}} \hypergeom \left (\left [\frac {1}{2}-\frac {a}{2}+\frac {n}{2}, \frac {1}{2}+\frac {a}{2}+\frac {n}{2}\right ], \left [\frac {3}{2}\right ], \frac {\cosh \left (2 x \right )}{2}+\frac {1}{2}\right ) \left (2 \cosh \left (2 x \right )-2\right )^{\frac {1}{4}} \left (\sinh ^{n}\relax (x )\right )}{\sqrt {\sinh \left (2 x \right )}} \]
✓ Solution by Mathematica
Time used: 0.532 (sec). Leaf size: 127
DSolve[y''[x] == -(Csch[x]^2*((1 - n)*n - a^2*Sinh[x]^2)*y[x]),y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {(-1)^{-n} \left (-\text {sech}^2(x)\right )^{a/2} \tanh ^2(x)^{\frac {1}{4}-\frac {n}{2}} \left (c_1 (-1)^n \tanh ^2(x)^n \, _2F_1\left (\frac {a+n}{2},\frac {1}{2} (a+n+1);n+\frac {1}{2};\tanh ^2(x)\right )+i c_2 \sqrt {\tanh ^2(x)} \, _2F_1\left (\frac {1}{2} (a-n+1),\frac {1}{2} (a-n+2);\frac {3}{2}-n;\tanh ^2(x)\right )\right )}{\sqrt {\tanh (x)}} \\ \end{align*}