Internal problem ID [9417]
Internal file name [OUTPUT/8357_Monday_June_06_2022_02_49_50_AM_62088145/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1088.
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.
\[ \boxed {4 y^{\prime \prime }+4 y^{\prime } \tan \left (x \right )-\left (5 \tan \left (x \right )^{2}+2\right ) y=0} \]
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 = arcsin(t)] Linear ODE actually solved: (3*t^2+2)*u(t)+(-4*t^4+8*t^2-4)*diff(diff(u(t),t),t) = 0 <- change of variables successful`
✓ Solution by Maple
Time used: 0.219 (sec). Leaf size: 31
dsolve(4*diff(diff(y(x),x),x)+4*diff(y(x),x)*tan(x)-(5*tan(x)^2+2)*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = \frac {i \cos \left (x \right ) \sin \left (x \right ) c_{2} -\ln \left (i \cos \left (x \right )+\sin \left (x \right )\right ) c_{2} +c_{1}}{\sqrt {\cos \left (x \right )}} \]
✓ Solution by Mathematica
Time used: 0.248 (sec). Leaf size: 97
DSolve[(-2 - 5*Tan[x]^2)*y[x] + 4*Tan[x]*y'[x] + 4*y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to \frac {3 (-1)^{7/8} c_2 \text {arcsinh}\left (\frac {(1+i) \sqrt [4]{-\cos ^4(x)}}{\sqrt {2}}\right )+3 \sqrt [8]{-1} c_2 \sqrt [4]{-\cos ^4(x)} \sqrt {1+i \sqrt {-\cos ^4(x)}}-2 (-1)^{7/8} c_1}{2 \sqrt [8]{-\cos ^4(x)}} \]