Internal problem ID [9358]
Internal file name [OUTPUT/8294_Monday_June_06_2022_02_39_27_AM_11422902/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1025.
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.
\[ \boxed {y^{\prime \prime }-\left (\frac {m \left (m -1\right )}{\cos \left (x \right )^{2}}+\frac {n \left (n -1\right )}{\sin \left (x \right )^{2}}+a \right ) y=0} \]
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 -> Whittaker -> 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 = arccos(t)] Linear ODE actually solved: (8*a*t^4+8*m^2*t^2-8*n^2*t^2-8*a*t^2-8*m*t^2+8*n*t^2-8*m^2+8*m)*u(t)+(8*t^5-8*t^3)*diff(u(t),t)+(8*t^6-16*t^4+8*t^2)*diff(diff <- change of variables successful`
✓ Solution by Maple
Time used: 0.375 (sec). Leaf size: 102
dsolve(diff(diff(y(x),x),x)-(m*(m-1)/cos(x)^2+n*(n-1)/sin(x)^2+a)*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = \sin \left (x \right )^{n} \left (c_{1} \cos \left (x \right )^{m} \operatorname {hypergeom}\left (\left [\frac {n}{2}+\frac {m}{2}+\frac {i \sqrt {a}}{2}, \frac {n}{2}+\frac {m}{2}-\frac {i \sqrt {a}}{2}\right ], \left [\frac {1}{2}+m \right ], \cos \left (x \right )^{2}\right )+c_{2} \cos \left (x \right )^{-m +1} \operatorname {hypergeom}\left (\left [\frac {n}{2}-\frac {m}{2}+\frac {i \sqrt {a}}{2}+\frac {1}{2}, \frac {n}{2}-\frac {m}{2}-\frac {i \sqrt {a}}{2}+\frac {1}{2}\right ], \left [\frac {3}{2}-m \right ], \cos \left (x \right )^{2}\right )\right ) \]
✓ Solution by Mathematica
Time used: 1.623 (sec). Leaf size: 158
DSolve[(-a - (-1 + n)*n*Csc[x]^2 - (-1 + m)*m*Sec[x]^2)*y[x] + y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to \frac {(-1)^{-m} \cos ^2(x)^{-\frac {m}{2}-\frac {1}{4}} \left (-\sin ^2(x)\right )^{n/2} \left (c_1 (-1)^m \cos ^2(x)^{m+\frac {1}{2}} \operatorname {Hypergeometric2F1}\left (\frac {1}{2} \left (m+n-\sqrt {-a}\right ),\frac {1}{2} \left (m+n+\sqrt {-a}\right ),m+\frac {1}{2},\cos ^2(x)\right )+i c_2 \cos ^2(x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2} \left (-m+n-\sqrt {-a}+1\right ),\frac {1}{2} \left (-m+n+\sqrt {-a}+1\right ),\frac {3}{2}-m,\cos ^2(x)\right )\right )}{\sqrt {\cos (x)}} \]