60.3.408 problem 1414

Internal problem ID [11418]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1414
Date solved : Tuesday, January 28, 2025 at 06:05:20 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }&=-\frac {\left (-a^{2} \sinh \left (x \right )^{2}-n \left (n -1\right )\right ) y}{\sinh \left (x \right )^{2}} \end{align*}

✓ Solution by Maple

Time used: 0.843 (sec). Leaf size: 82

dsolve(diff(diff(y(x),x),x) = -(-a^2*sinh(x)^2-n*(n-1))/sinh(x)^2*y(x),y(x), singsol=all)
 

\[ y = \frac {\sqrt {\cosh \left (x \right )}\, \sinh \left (x \right )^{n +\frac {1}{2}} \left (\cosh \left (x \right ) \operatorname {hypergeom}\left (\left [\frac {1}{2}+\frac {a}{2}+\frac {n}{2}, \frac {1}{2}-\frac {a}{2}+\frac {n}{2}\right ], \left [\frac {3}{2}\right ], \frac {\cosh \left (2 x \right )}{2}+\frac {1}{2}\right ) c_{2} +\operatorname {hypergeom}\left (\left [-\frac {a}{2}+\frac {n}{2}, \frac {a}{2}+\frac {n}{2}\right ], \left [\frac {1}{2}\right ], \frac {\cosh \left (2 x \right )}{2}+\frac {1}{2}\right ) c_{1} \right )}{\sqrt {\sinh \left (2 x \right )}} \]

✓ Solution by Mathematica

Time used: 1.037 (sec). Leaf size: 127

DSolve[D[y[x],{x,2}] == -(Csch[x]^2*((1 - n)*n - a^2*Sinh[x]^2)*y[x]),y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to \frac {(-1)^{-n} \left (-\text {sech}^2(x)\right )^{a/2} \tanh ^2(x)^{-\frac {n}{2}-\frac {1}{4}} \left (c_1 (-1)^n \tanh ^2(x)^{n+\frac {1}{2}} \operatorname {Hypergeometric2F1}\left (\frac {a+n}{2},\frac {1}{2} (a+n+1),n+\frac {1}{2},\tanh ^2(x)\right )+i c_2 \tanh ^2(x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2} (a-n+1),\frac {1}{2} (a-n+2),\frac {3}{2}-n,\tanh ^2(x)\right )\right )}{\sqrt {\tanh (x)}} \]