Internal problem ID [9747]
Internal file name [OUTPUT/8689_Monday_June_06_2022_05_13_00_AM_66531319/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1420.
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.
\[ \boxed {\cos \left (x \right )^{2} y^{\prime \prime }-\left (a \cos \left (x \right )^{2}+n \left (n -1\right )\right ) y=0} \]
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 -> 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*arccos(t)] Linear ODE actually solved: (-a*t-2*n^2-a+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: 0.609 (sec). Leaf size: 114
dsolve(cos(x)^2*diff(diff(y(x),x),x)-(a*cos(x)^2+n*(n-1))*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = \frac {\sin \left (x \right )^{\frac {3}{2}} \left (\operatorname {hypergeom}\left (\left [\frac {1}{2}+\frac {i \sqrt {a}}{2}+\frac {n}{2}, \frac {1}{2}-\frac {i \sqrt {a}}{2}+\frac {n}{2}\right ], \left [n +\frac {1}{2}\right ], \frac {\cos \left (2 x \right )}{2}+\frac {1}{2}\right ) \cos \left (x \right )^{n +\frac {1}{2}} c_{1} +\left (\frac {\cos \left (2 x \right )}{2}+\frac {1}{2}\right )^{\frac {3}{4}-\frac {n}{2}} \operatorname {hypergeom}\left (\left [1+\frac {i \sqrt {a}}{2}-\frac {n}{2}, 1-\frac {i \sqrt {a}}{2}-\frac {n}{2}\right ], \left [\frac {3}{2}-n \right ], \frac {\cos \left (2 x \right )}{2}+\frac {1}{2}\right ) c_{2} \right )}{\sqrt {\sin \left (2 x \right )}} \]
✓ Solution by Mathematica
Time used: 0.598 (sec). Leaf size: 126
DSolve[(-((-1 + n)*n) - a*Cos[x]^2)*y[x] + Cos[x]^2*y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to c_1 i^{1-n} \cos ^{1-n}(x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2} \left (-n-i \sqrt {a}+1\right ),\frac {1}{2} \left (-n+i \sqrt {a}+1\right ),\frac {3}{2}-n,\cos ^2(x)\right )+c_2 i^n \cos ^n(x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2} \left (n-i \sqrt {a}\right ),\frac {1}{2} \left (n+i \sqrt {a}\right ),n+\frac {1}{2},\cos ^2(x)\right ) \]