Internal problem ID [9755]
Internal file name [OUTPUT/8697_Monday_June_06_2022_05_14_20_AM_42527363/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1428.
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.
\[ \boxed {y^{\prime \prime }+\frac {\left (a \cos \left (x \right )^{2}+b \sin \left (x \right )^{2}+c \right ) y}{\sin \left (x \right )^{2}}=0} \]
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-b*t+a+b+2*c)*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: 0.594 (sec). Leaf size: 161
dsolve(diff(diff(y(x),x),x) = -(a*cos(x)^2+b*sin(x)^2+c)/sin(x)^2*y(x),y(x), singsol=all)
\[ y \left (x \right ) = \frac {\sqrt {\cos \left (x \right )}\, \left (-\frac {1}{2}+\frac {\cos \left (2 x \right )}{2}\right )^{\frac {1}{2}+\frac {\sqrt {-4 a +1-4 c}}{4}} \left (\operatorname {hypergeom}\left (\left [\frac {\sqrt {-4 a +1-4 c}}{4}+\frac {\sqrt {-a +b}}{2}+\frac {3}{4}, \frac {\sqrt {-4 a +1-4 c}}{4}-\frac {\sqrt {-a +b}}{2}+\frac {3}{4}\right ], \left [\frac {3}{2}\right ], \frac {\cos \left (2 x \right )}{2}+\frac {1}{2}\right ) \cos \left (x \right ) c_{2} +\operatorname {hypergeom}\left (\left [\frac {\sqrt {-4 a +1-4 c}}{4}-\frac {\sqrt {-a +b}}{2}+\frac {1}{4}, \frac {\sqrt {-4 a +1-4 c}}{4}+\frac {\sqrt {-a +b}}{2}+\frac {1}{4}\right ], \left [\frac {1}{2}\right ], \frac {\cos \left (2 x \right )}{2}+\frac {1}{2}\right ) c_{1} \right )}{\sqrt {\sin \left (2 x \right )}} \]
✓ Solution by Mathematica
Time used: 0.567 (sec). Leaf size: 87
DSolve[y''[x] == -(Csc[x]^2*(c + a*Cos[x]^2 + b*Sin[x]^2)*y[x]),y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to \sqrt [4]{-\sin ^2(x)} \left (c_1 P_{\sqrt {b-a}-\frac {1}{2}}^{\frac {1}{2} \sqrt {-4 a-4 c+1}}(\cos (x))+c_2 Q_{\sqrt {b-a}-\frac {1}{2}}^{\frac {1}{2} \sqrt {-4 a-4 c+1}}(\cos (x))\right ) \]