Internal problem ID [9896]
Internal file name [OUTPUT/8843_Monday_June_06_2022_05_36_55_AM_82946051/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 4, linear fourth order
Problem number: 1574.
ODE order: 4.
ODE degree: 1.
The type(s) of ODE detected by this program : "unknown"
Maple gives the following as the ode type
[[_high_order, _with_linear_symmetries]]
Unable to solve or complete the solution.
\[ \boxed {y^{\prime \prime \prime \prime } \sin \left (x \right )^{4}+2 y^{\prime \prime \prime } \sin \left (x \right )^{3} \cos \left (x \right )+y^{\prime \prime } \sin \left (x \right )^{2} \left (\sin \left (x \right )^{2}-3\right )+y^{\prime } \sin \left (x \right ) \cos \left (x \right ) \left (2 \sin \left (x \right )^{2}+3\right )+\left (a^{4} \sin \left (x \right )^{4}-3\right ) y=0} \] Unable to solve this ODE.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \left (\frac {d}{d x}\frac {d}{d x}y^{\prime \prime }\right ) \sin \left (x \right )^{4}+2 \left (\frac {d}{d x}y^{\prime \prime }\right ) \sin \left (x \right )^{3} \cos \left (x \right )+\left (\frac {d}{d x}y^{\prime }\right ) \sin \left (x \right )^{2} \left (\sin \left (x \right )^{2}-3\right )+y^{\prime } \sin \left (x \right ) \cos \left (x \right ) \left (2 \sin \left (x \right )^{2}+3\right )+\left (a^{4} \sin \left (x \right )^{4}-3\right ) y=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 4 \\ {} & {} & \frac {d}{d x}\frac {d}{d x}y^{\prime \prime } \end {array} \]
Maple trace
`Methods for high order ODEs: --- Trying classification methods --- trying a quadrature checking if the LODE has constant coefficients checking if the LODE is of Euler type trying high order exact linear fully integrable trying to convert to a linear ODE with constant coefficients trying differential order: 4; missing the dependent variable trying a solution in terms of MeijerG functions -> Try computing a Rational Normal Form for the given ODE... <- unable to resolve the Equivalence to a Rational Normal Form trying differential order: 4; missing the dependent variable Multiplying solutions by`, exp(Int(z/((z-1)*(z+1)), z))`Equation is the LCLM of ((-a^4+1)^(1/2)*z^2-z^2-(-a^4+1)^(1/2))/(z-1)^2/(z+1 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 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 <- solving the LCLM ode successful substitution methods successful`
✓ Solution by Maple
Time used: 0.688 (sec). Leaf size: 204
dsolve(diff(diff(diff(diff(y(x),x),x),x),x)*sin(x)^4+2*diff(diff(diff(y(x),x),x),x)*sin(x)^3*cos(x)+diff(diff(y(x),x),x)*sin(x)^2*(sin(x)^2-3)+diff(y(x),x)*sin(x)*cos(x)*(2*sin(x)^2+3)+(a^4*sin(x)^4-3)*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = \sin \left (x \right ) \left (c_{1} \operatorname {hypergeom}\left (\left [\frac {3}{4}-\frac {\sqrt {-4 \sqrt {-a^{4}+1}+5}}{4}, \frac {3}{4}+\frac {\sqrt {-4 \sqrt {-a^{4}+1}+5}}{4}\right ], \left [\frac {1}{2}\right ], \cos \left (x \right )^{2}\right )+c_{2} \operatorname {hypergeom}\left (\left [\frac {3}{4}-\frac {\sqrt {4 \sqrt {-a^{4}+1}+5}}{4}, \frac {3}{4}+\frac {\sqrt {4 \sqrt {-a^{4}+1}+5}}{4}\right ], \left [\frac {1}{2}\right ], \cos \left (x \right )^{2}\right )+\cos \left (x \right ) \left (\operatorname {hypergeom}\left (\left [\frac {5}{4}+\frac {\sqrt {-4 \sqrt {-a^{4}+1}+5}}{4}, \frac {5}{4}-\frac {\sqrt {-4 \sqrt {-a^{4}+1}+5}}{4}\right ], \left [\frac {3}{2}\right ], \cos \left (x \right )^{2}\right ) c_{3} +c_{4} \operatorname {hypergeom}\left (\left [\frac {5}{4}+\frac {\sqrt {4 \sqrt {-a^{4}+1}+5}}{4}, \frac {5}{4}-\frac {\sqrt {4 \sqrt {-a^{4}+1}+5}}{4}\right ], \left [\frac {3}{2}\right ], \cos \left (x \right )^{2}\right )\right )\right ) \]
✓ Solution by Mathematica
Time used: 0.232 (sec). Leaf size: 265
DSolve[(-3 + a^4*Sin[x]^4)*y[x] + Cos[x]*Sin[x]*(3 + 2*Sin[x]^2)*y'[x] + Sin[x]^2*(-3 + Sin[x]^2)*y''[x] + 2*Cos[x]*Sin[x]^3*Derivative[3][y][x] + Sin[x]^4*Derivative[4][y][x] == 0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to \sin (x) \left (c_1 \operatorname {Hypergeometric2F1}\left (\frac {1}{4} \left (3-\sqrt {5-4 \sqrt {1-a^4}}\right ),\frac {1}{4} \left (\sqrt {5-4 \sqrt {1-a^4}}+3\right ),\frac {1}{2},\cos ^2(x)\right )+c_3 \cos (x) \operatorname {Hypergeometric2F1}\left (\frac {1}{4} \left (5-\sqrt {5-4 \sqrt {1-a^4}}\right ),\frac {1}{4} \left (\sqrt {5-4 \sqrt {1-a^4}}+5\right ),\frac {3}{2},\cos ^2(x)\right )+c_2 \operatorname {Hypergeometric2F1}\left (\frac {1}{4} \left (3-\sqrt {4 \sqrt {1-a^4}+5}\right ),\frac {1}{4} \left (\sqrt {4 \sqrt {1-a^4}+5}+3\right ),\frac {1}{2},\cos ^2(x)\right )+c_4 \cos (x) \operatorname {Hypergeometric2F1}\left (\frac {1}{4} \left (5-\sqrt {4 \sqrt {1-a^4}+5}\right ),\frac {1}{4} \left (\sqrt {4 \sqrt {1-a^4}+5}+5\right ),\frac {3}{2},\cos ^2(x)\right )\right ) \]