Internal problem ID [8994]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1415.
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 n \cosh \relax (x ) y^{\prime }}{\sinh \relax (x )}+\left (-a^{2}+n^{2}\right ) y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.532 (sec). Leaf size: 43
dsolve(diff(diff(y(x),x),x) = -2*n/sinh(x)*cosh(x)*diff(y(x),x)-(-a^2+n^2)*y(x),y(x), singsol=all)
\[ y \relax (x ) = c_{1} \left (\sinh ^{\frac {1}{2}-n}\relax (x )\right ) \LegendreP \left (a -\frac {1}{2}, n -\frac {1}{2}, \cosh \relax (x )\right )+c_{2} \left (\sinh ^{\frac {1}{2}-n}\relax (x )\right ) \LegendreQ \left (a -\frac {1}{2}, n -\frac {1}{2}, \cosh \relax (x )\right ) \]
✓ Solution by Mathematica
Time used: 0.454 (sec). Leaf size: 145
DSolve[y''[x] == (a^2 - n^2)*y[x] - 2*n*Coth[x]*y'[x],y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to (-1)^{-n} \left (-\text {sech}^2(x)\right )^{\frac {a+1}{2}} \tanh ^{-n-\frac {1}{2}}(x) \tanh ^2(x)^{-\frac {n}{2}-\frac {1}{4}} \text {sech}^2(x)^{\frac {n-1}{2}} \left (c_1 (-1)^n \tanh ^2(x)^{n+\frac {1}{2}} \, _2F_1\left (\frac {a+n}{2},\frac {1}{2} (a+n+1);n+\frac {1}{2};\tanh ^2(x)\right )+i c_2 \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 ) \\ \end{align*}