Internal problem ID [9357]
Internal file name [OUTPUT/8293_Monday_June_06_2022_02_39_21_AM_84902345/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1024.
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.
\[ \boxed {y^{\prime \prime }-\left (1+2 \tan \left (x \right )^{2}\right ) y=0} \]
Maple trace Kovacic algorithm successful
`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 A Liouvillian solution exists Reducible group (found an exponential solution) Group is reducible, not completely reducible <- Kovacics algorithm successful Change of variables used: [x = arccos(t)] Linear ODE actually solved: (t^2-2)*u(t)-t^3*diff(u(t),t)+(-t^4+t^2)*diff(diff(u(t),t),t) = 0 <- change of variables successful`
✓ Solution by Maple
Time used: 0.172 (sec). Leaf size: 28
dsolve(diff(diff(y(x),x),x)-(1+2*tan(x)^2)*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = i \sin \left (x \right ) c_{2} +\sec \left (x \right ) \ln \left (\cos \left (x \right )+i \sin \left (x \right )\right ) c_{2} +c_{1} \sec \left (x \right ) \]
✓ Solution by Mathematica
Time used: 0.501 (sec). Leaf size: 46
DSolve[(-1 - 2*Tan[x]^2)*y[x] + y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to c_2 \sec (x) \arctan \left (\frac {\cos (x)}{\sqrt {\sin ^2(x)}-1}\right )-\frac {1}{2} c_2 \sqrt {\sin ^2(x)}+c_1 \sec (x) \]