Internal problem ID [9742]
Internal file name [OUTPUT/8684_Monday_June_06_2022_05_11_18_AM_29221059/index.tex]
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.
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.
\[ \boxed {y^{\prime \prime }+\frac {2 n \cosh \left (x \right ) y^{\prime }}{\sinh \left (x \right )}+\left (-a^{2}+n^{2}\right ) y=0} \]
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 <- Legendre successful <- special function solution successful Change of variables used: [x = arccosh(t)] Linear ODE actually solved: (-a^2*t^2+n^2*t^2+a^2-n^2)*u(t)+(2*n*t^3+t^3-2*n*t-t)*diff(u(t),t)+(t^4-2*t^2+1)*diff(diff(u(t),t),t) = 0 <- change of variables successful`
✓ Solution by Maple
Time used: 0.234 (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)
\[ y \left (x \right ) = \sinh \left (x \right )^{-n +\frac {1}{2}} \left (c_{1} \operatorname {LegendreP}\left (a -\frac {1}{2}, n -\frac {1}{2}, \cosh \left (x \right )\right )+c_{2} \operatorname {LegendreQ}\left (a -\frac {1}{2}, n -\frac {1}{2}, \cosh \left (x \right )\right )\right ) \]
✓ Solution by Mathematica
Time used: 1.034 (sec). Leaf size: 145
DSolve[y''[x] == (a^2 - n^2)*y[x] - 2*n*Coth[x]*y'[x],y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to (-1)^{-n} \left (-\text {sech}^2(x)\right )^{\frac {a+1}{2}} \tanh ^{-n-\frac {1}{2}}(x) \tanh ^2(x)^{-\frac {n}{2}-\frac {1}{4}} \text {sech}^2(x)^{\frac {n-1}{2}} \left (c_1 (-1)^n \tanh ^2(x)^{n+\frac {1}{2}} \operatorname {Hypergeometric2F1}\left (\frac {a+n}{2},\frac {1}{2} (a+n+1),n+\frac {1}{2},\tanh ^2(x)\right )+i c_2 \tanh ^2(x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2} (a-n+1),\frac {1}{2} (a-n+2),\frac {3}{2}-n,\tanh ^2(x)\right )\right ) \]