Internal problem ID [9753]
Internal file name [OUTPUT/8695_Monday_June_06_2022_05_13_59_AM_75946640/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1426.
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.
\[ \boxed {y^{\prime \prime } \sin \left (x \right )^{2}-\left (a^{2} \cos \left (x \right )^{2}+b \cos \left (x \right )+\frac {b^{2}}{\left (2 a -3\right )^{2}}+3 a +2\right ) y=0} \]
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 -> Whittaker -> 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 = arccos(t)] Linear ODE actually solved: (-8*a^4*t^2+24*a^3*t^2-8*a^2*b*t-18*a^2*t^2-24*a^3+24*a*b*t+56*a^2-2*b^2-18*b*t-6*a-36)*u(t)+(8*a^2*t^3-24*a*t^3-8*a^2*t+18*t^ <- change of variables successful`
✓ Solution by Maple
Time used: 0.719 (sec). Leaf size: 558
dsolve(sin(x)^2*diff(diff(y(x),x),x)-(a^2*cos(x)^2+b*cos(x)+b^2/(2*a-3)^2+3*a+2)*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = \frac {\left (\frac {\cos \left (x \right )}{2}-\frac {1}{2}\right )^{\frac {-6+4 a +\sqrt {16 a^{4}+\left (16 b -72\right ) a^{2}-48 a b +4 \left (\frac {9}{2}+b \right )^{2}}}{-12+8 a}} \left (\cos \left (\frac {x}{2}\right )^{\frac {-6+4 a +\sqrt {16 a^{4}+\left (-16 b -72\right ) a^{2}+48 a b +4 \left (b -\frac {9}{2}\right )^{2}}}{-6+4 a}} \operatorname {hypergeom}\left (\left [\frac {8 a^{2}+\sqrt {16 a^{4}+\left (-16 b -72\right ) a^{2}+48 a b +4 \left (b -\frac {9}{2}\right )^{2}}+\sqrt {16 a^{4}+\left (16 b -72\right ) a^{2}-48 a b +4 \left (\frac {9}{2}+b \right )^{2}}-8 a -6}{-12+8 a}, \frac {-8 a^{2}+\sqrt {16 a^{4}+\left (-16 b -72\right ) a^{2}+48 a b +4 \left (b -\frac {9}{2}\right )^{2}}+\sqrt {16 a^{4}+\left (16 b -72\right ) a^{2}-48 a b +4 \left (\frac {9}{2}+b \right )^{2}}+16 a -6}{-12+8 a}\right ], \left [\frac {-6+4 a +\sqrt {16 a^{4}+\left (-16 b -72\right ) a^{2}+48 a b +4 \left (b -\frac {9}{2}\right )^{2}}}{-6+4 a}\right ], \frac {\cos \left (x \right )}{2}+\frac {1}{2}\right ) c_{2} +\cos \left (\frac {x}{2}\right )^{\frac {-6+4 a -\sqrt {16 a^{4}+\left (-16 b -72\right ) a^{2}+48 a b +4 \left (b -\frac {9}{2}\right )^{2}}}{-6+4 a}} \operatorname {hypergeom}\left (\left [\frac {8 a^{2}-\sqrt {16 a^{4}+\left (-16 b -72\right ) a^{2}+48 a b +4 \left (b -\frac {9}{2}\right )^{2}}+\sqrt {16 a^{4}+\left (16 b -72\right ) a^{2}-48 a b +4 \left (\frac {9}{2}+b \right )^{2}}-8 a -6}{-12+8 a}, \frac {-8 a^{2}-\sqrt {16 a^{4}+\left (-16 b -72\right ) a^{2}+48 a b +4 \left (b -\frac {9}{2}\right )^{2}}+\sqrt {16 a^{4}+\left (16 b -72\right ) a^{2}-48 a b +4 \left (\frac {9}{2}+b \right )^{2}}+16 a -6}{-12+8 a}\right ], \left [\frac {-6+4 a -\sqrt {16 a^{4}+\left (-16 b -72\right ) a^{2}+48 a b +4 \left (b -\frac {9}{2}\right )^{2}}}{-6+4 a}\right ], \frac {\cos \left (x \right )}{2}+\frac {1}{2}\right ) c_{1} \right )}{\sqrt {\sin \left (x \right )}} \]
✓ Solution by Mathematica
Time used: 6.668 (sec). Leaf size: 1281
DSolve[(-2 - 3*a - b^2/(-3 + 2*a)^2 - b*Cos[x] - a^2*Cos[x]^2)*y[x] + Sin[x]^2*y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to \frac {(-1)^{\frac {-4 a^2-9}{(3-2 a)^2}} 2^{-\frac {\sqrt {(3-2 a)^2 \left (16 a^4+8 (2 b-9) a^2-48 b a+(2 b+9)^2\right )}}{2 (3-2 a)^2}} (\cos (x)-1)^{-\frac {-8 a^2+24 a+\sqrt {(3-2 a)^2 \left (16 a^4+8 (2 b-9) a^2-48 b a+(2 b+9)^2\right )}-18}{4 (3-2 a)^2}} (\cos (x)+1)^{\frac {1}{4} \left (\sqrt {\frac {16 a^4-8 (2 b+9) a^2+48 b a+(9-2 b)^2}{(3-2 a)^2}}+2\right )} \left ((-1)^{\frac {4 a^2+9}{(3-2 a)^2}} 2^{\frac {\sqrt {(3-2 a)^2 \left (16 a^4+8 (2 b-9) a^2-48 b a+(2 b+9)^2\right )}}{2 (3-2 a)^2}} c_1 \operatorname {Hypergeometric2F1}\left (\frac {16 a^3+4 \left (\sqrt {\frac {16 a^4-8 (2 b+9) a^2+48 b a+(9-2 b)^2}{(3-2 a)^2}}-10\right ) a^2-12 \left (\sqrt {\frac {16 a^4-8 (2 b+9) a^2+48 b a+(9-2 b)^2}{(3-2 a)^2}}-1\right ) a+9 \sqrt {\frac {16 a^4-8 (2 b+9) a^2+48 b a+(9-2 b)^2}{(3-2 a)^2}}-\sqrt {(3-2 a)^2 \left (16 a^4+8 (2 b-9) a^2-48 b a+(2 b+9)^2\right )}+18}{4 (3-2 a)^2},-\frac {16 a^3-4 \left (\sqrt {\frac {16 a^4-8 (2 b+9) a^2+48 b a+(9-2 b)^2}{(3-2 a)^2}}+14\right ) a^2+12 \left (\sqrt {\frac {16 a^4-8 (2 b+9) a^2+48 b a+(9-2 b)^2}{(3-2 a)^2}}+5\right ) a-9 \sqrt {\frac {16 a^4-8 (2 b+9) a^2+48 b a+(9-2 b)^2}{(3-2 a)^2}}+\sqrt {(3-2 a)^2 \left (16 a^4+8 (2 b-9) a^2-48 b a+(2 b+9)^2\right )}-18}{4 (3-2 a)^2},-\frac {-8 a^2+24 a+\sqrt {(3-2 a)^2 \left (16 a^4+8 (2 b-9) a^2-48 b a+(2 b+9)^2\right )}-18}{2 (3-2 a)^2},\sin ^2\left (\frac {x}{2}\right )\right )-i^{\frac {24 a+\sqrt {(3-2 a)^2 \left (16 a^4+8 (2 b-9) a^2-48 b a+(2 b+9)^2\right )}}{(3-2 a)^2}} c_2 (1-\cos (x))^{\frac {\sqrt {(3-2 a)^2 \left (16 a^4+8 (2 b-9) a^2-48 b a+(2 b+9)^2\right )}}{2 (3-2 a)^2}} \operatorname {Hypergeometric2F1}\left (\frac {16 a^3+4 \left (\sqrt {\frac {16 a^4-8 (2 b+9) a^2+48 b a+(9-2 b)^2}{(3-2 a)^2}}-10\right ) a^2-12 \left (\sqrt {\frac {16 a^4-8 (2 b+9) a^2+48 b a+(9-2 b)^2}{(3-2 a)^2}}-1\right ) a+9 \sqrt {\frac {16 a^4-8 (2 b+9) a^2+48 b a+(9-2 b)^2}{(3-2 a)^2}}+\sqrt {(3-2 a)^2 \left (16 a^4+8 (2 b-9) a^2-48 b a+(2 b+9)^2\right )}+18}{4 (3-2 a)^2},\frac {-16 a^3+4 \left (\sqrt {\frac {16 a^4-8 (2 b+9) a^2+48 b a+(9-2 b)^2}{(3-2 a)^2}}+14\right ) a^2-12 \left (\sqrt {\frac {16 a^4-8 (2 b+9) a^2+48 b a+(9-2 b)^2}{(3-2 a)^2}}+5\right ) a+9 \sqrt {\frac {16 a^4-8 (2 b+9) a^2+48 b a+(9-2 b)^2}{(3-2 a)^2}}+\sqrt {(3-2 a)^2 \left (16 a^4+8 (2 b-9) a^2-48 b a+(2 b+9)^2\right )}+18}{4 (3-2 a)^2},\frac {8 a^2-24 a+\sqrt {(3-2 a)^2 \left (16 a^4+8 (2 b-9) a^2-48 b a+(2 b+9)^2\right )}+18}{2 (3-2 a)^2},\sin ^2\left (\frac {x}{2}\right )\right )\right )}{\sqrt [4]{-\sin ^2(x)}} \]