60.3.409 problem 1415

Internal problem ID [11419]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1415
Date solved : Tuesday, January 28, 2025 at 06:05:21 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Solution by Maple

Time used: 0.464 (sec). Leaf size: 36

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)
 

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

✓ Solution by Mathematica

Time used: 1.505 (sec). Leaf size: 149

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

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