Internal problem ID [9743]
Internal file name [OUTPUT/8685_Monday_June_06_2022_05_11_34_AM_89973403/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1416.
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.
\[ \boxed {y^{\prime \prime }+\frac {\left (1+2 n \right ) \cos \left (x \right ) y^{\prime }}{\sin \left (x \right )}+\left (v +n +1\right ) \left (v -n \right ) y=0} \]
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 = arccos(t)] Linear ODE actually solved: (n^2*t^2-t^2*v^2+n*t^2-t^2*v-n^2+v^2-n+v)*u(t)+(2*n*t^3+2*t^3-2*n*t-2*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: 5.375 (sec). Leaf size: 26
dsolve(diff(diff(y(x),x),x) = -(2*n+1)*cos(x)/sin(x)*diff(y(x),x)-(v+n+1)*(v-n)*y(x),y(x), singsol=all)
\[ y \left (x \right ) = \sin \left (x \right )^{-n} \left (c_{1} \operatorname {LegendreP}\left (v , n , \cos \left (x \right )\right )+c_{2} \operatorname {LegendreQ}\left (v , n , \cos \left (x \right )\right )\right ) \]
✓ Solution by Mathematica
Time used: 0.236 (sec). Leaf size: 35
DSolve[y''[x] == (n - v)*(1 + n + v)*y[x] - (1 + 2*n)*Cot[x]*y'[x],y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to \left (-\sin ^2(x)\right )^{-n/2} (c_1 P_v^n(\cos (x))+c_2 Q_v^n(\cos (x))) \]