Internal problem ID [9397]
Internal file name [OUTPUT/8334_Monday_June_06_2022_02_45_53_AM_42635226/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1065.
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.
\[ \boxed {y^{\prime \prime }+2 n y^{\prime } \cot \left (x \right )+\left (-a^{2}+n^{2}\right ) y=0} \]
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 -> Kummer -> hyper3: Equivalence to 1F1 under a power @ Moebius -> hypergeometric -> heuristic approach <- heuristic approach successful <- hypergeometric successful <- special function solution successful Change of variables used: [x = arcsin(t)] Linear ODE actually solved: (-a^2*t+n^2*t)*u(t)+(-2*n*t^2-t^2+2*n)*diff(u(t),t)+(-t^3+t)*diff(diff(u(t),t),t) = 0 <- change of variables successful`
✓ Solution by Maple
Time used: 0.359 (sec). Leaf size: 60
dsolve(diff(diff(y(x),x),x)+2*n*diff(y(x),x)*cot(x)+(-a^2+n^2)*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = \sin \left (x \right )^{-n +\frac {1}{2}} \left (c_{1} \operatorname {LegendreP}\left (-\frac {1}{2}+\sqrt {-a^{2}+2 n^{2}}, n -\frac {1}{2}, \cos \left (x \right )\right )+c_{2} \operatorname {LegendreQ}\left (-\frac {1}{2}+\sqrt {-a^{2}+2 n^{2}}, n -\frac {1}{2}, \cos \left (x \right )\right )\right ) \]
✓ Solution by Mathematica
Time used: 0.247 (sec). Leaf size: 83
DSolve[(-a^2 + n^2)*y[x] + 2*n*Cot[x]*y'[x] + y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to \left (-\sin ^2(x)\right )^{\frac {1}{4}-\frac {n}{2}} \left (c_1 P_{\sqrt {2 n^2-a^2}-\frac {1}{2}}^{n-\frac {1}{2}}(\cos (x))+c_2 Q_{\sqrt {2 n^2-a^2}-\frac {1}{2}}^{n-\frac {1}{2}}(\cos (x))\right ) \]