Internal problem ID [9741]
Internal file name [OUTPUT/8683_Monday_June_06_2022_05_11_07_AM_25832827/index.tex]
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.
The type(s) of ODE detected by this program : "unknown"
Maple gives the following as the ode type
[[_2nd_order, _with_linear_symmetries]]
Unable to solve or complete the solution.
\[ \boxed {y^{\prime \prime }+\frac {\left (-a^{2} \sinh \left (x \right )^{2}-n \left (n -1\right )\right ) y}{\sinh \left (x \right )^{2}}=0} \]
Maple trace
`Methods for second order ODEs: --- Trying classification methods --- trying a symmetry of the form [xi=0, eta=F(x)] checking if the LODE is missing y -> Heun: Equivalence to the GHE or one of its 4 confluent cases under a power @ Moebius -> trying a solution of the form r0(x) * Y + r1(x) * Y where Y = exp(int(r(x), dx)) * 2F1([a1, a2], [b1], f) -> Trying changes of variables to rationalize or make the ODE simpler trying a quadrature checking if the LODE has constant coefficients checking if the LODE is of Euler type trying a symmetry of the form [xi=0, eta=F(x)] checking if the LODE is missing y -> Trying a Liouvillian solution using Kovacics algorithm <- No Liouvillian solutions exists -> Trying a solution in terms of special functions: -> Bessel -> elliptic -> Legendre -> Kummer -> hyper3: Equivalence to 1F1 under a power @ Moebius -> hypergeometric -> heuristic approach -> hyper3: Equivalence to 2F1, 1F1 or 0F1 under a power @ Moebius <- hyper3 successful: received ODE is equivalent to the 2F1 ODE <- hypergeometric successful <- special function solution successful Change of variables used: [x = 1/2*arccosh(t)] Linear ODE actually solved: (-a^2*t+a^2-2*n^2+2*n)*u(t)+(4*t^2-4*t)*diff(u(t),t)+(4*t^3-4*t^2-4*t+4)*diff(diff(u(t),t),t) = 0 <- change of variables successful`
✓ Solution by Maple
Time used: 0.578 (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)
\[ y \left (x \right ) = \frac {\sinh \left (x \right )^{n +\frac {1}{2}} \sqrt {\cosh \left (x \right )}\, \left (\operatorname {hypergeom}\left (\left [\frac {1}{2}-\frac {a}{2}+\frac {n}{2}, \frac {1}{2}+\frac {a}{2}+\frac {n}{2}\right ], \left [\frac {3}{2}\right ], \frac {\cosh \left (2 x \right )}{2}+\frac {1}{2}\right ) \cosh \left (x \right ) c_{2} +\operatorname {hypergeom}\left (\left [-\frac {a}{2}+\frac {n}{2}, \frac {a}{2}+\frac {n}{2}\right ], \left [\frac {1}{2}\right ], \frac {\cosh \left (2 x \right )}{2}+\frac {1}{2}\right ) c_{1} \right )}{\sqrt {\sinh \left (2 x \right )}} \]
✓ Solution by Mathematica
Time used: 1.203 (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]
\[ y(x)\to \frac {(-1)^{-n} \left (-\text {sech}^2(x)\right )^{a/2} \tanh ^2(x)^{-\frac {n}{2}-\frac {1}{4}} \left (c_1 (-1)^n \tanh ^2(x)^{n+\frac {1}{2}} \operatorname {Hypergeometric2F1}\left (\frac {a+n}{2},\frac {1}{2} (a+n+1),n+\frac {1}{2},\tanh ^2(x)\right )+i c_2 \tanh ^2(x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2} (a-n+1),\frac {1}{2} (a-n+2),\frac {3}{2}-n,\tanh ^2(x)\right )\right )}{\sqrt {\tanh (x)}} \]