Internal problem ID [5828]
Internal file name [OUTPUT/5076_Sunday_June_05_2022_03_20_25_PM_94345149/index.tex]
Book: Ordinary differential equations and calculus of variations. Makarets and Reshetnyak. Wold
Scientific. Singapore. 1995
Section: Chapter 2. Linear homogeneous equations. Section 2.3.4 problems. page 104
Problem number: 7.
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, _linear, _nonhomogeneous]]
Unable to solve or complete the solution.
\[ \boxed {x^{2} \left (x +1\right ) y^{\prime \prime }+x \left (4 x +3\right ) y^{\prime }-y=x +\frac {1}{x}} \]
Maple trace
`Methods for second order ODEs: --- Trying classification methods --- trying a quadrature trying high order exact linear fully integrable trying differential order: 2; linear nonhomogeneous with symmetry [0,1] trying a double symmetry of the form [xi=0, eta=F(x)] -> Try solving first the homogeneous part of the ODE 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 <- solving first the homogeneous part of the ODE successful`
✓ Solution by Maple
Time used: 0.141 (sec). Leaf size: 640
dsolve(x^2*(x+1)*diff(y(x),x$2)+x*(4*x+3)*diff(y(x),x)-y(x)=x+1/x,y(x), singsol=all)
\[ y \left (x \right ) = \frac {-5 x^{-\sqrt {2}} \left (\sqrt {2}-\frac {6}{5}\right ) \operatorname {hypergeom}\left (\left [2-\sqrt {2}, -1-\sqrt {2}\right ], \left [1-2 \sqrt {2}\right ], -x \right ) \left (\int \frac {x^{\sqrt {2}-1} \left (\left (\left (-2 x^{2}-4 x -\frac {5}{2}\right ) \sqrt {2}-x^{2}-\frac {11 x}{2}-3\right ) \operatorname {hypergeom}\left (\left [\sqrt {2}-1, \sqrt {2}-1\right ], \left [1+2 \sqrt {2}\right ], -x \right )+\left (\sqrt {2}-1\right ) x \left (x^{2}+\frac {1}{2} x +\frac {1}{2}\right ) \operatorname {hypergeom}\left (\left [\sqrt {2}, \sqrt {2}\right ], \left [2+2 \sqrt {2}\right ], -x \right )\right ) \left (x^{2}+1\right )}{\left (-7 \sqrt {2}\, \operatorname {hypergeom}\left (\left [\sqrt {2}-1, \sqrt {2}-1\right ], \left [1+2 \sqrt {2}\right ], -x \right )+4 x \operatorname {hypergeom}\left (\left [\sqrt {2}, \sqrt {2}\right ], \left [2+2 \sqrt {2}\right ], -x \right ) \left (\sqrt {2}-\frac {11}{8}\right )\right ) \operatorname {hypergeom}\left (\left [-1-\sqrt {2}, -1-\sqrt {2}\right ], \left [1-2 \sqrt {2}\right ], -x \right )+4 x \operatorname {hypergeom}\left (\left [-\sqrt {2}, -\sqrt {2}\right ], \left [2-2 \sqrt {2}\right ], -x \right ) \operatorname {hypergeom}\left (\left [\sqrt {2}-1, \sqrt {2}-1\right ], \left [1+2 \sqrt {2}\right ], -x \right ) \left (\sqrt {2}+\frac {11}{8}\right )}d x \right )+5 x^{\sqrt {2}} \left (\sqrt {2}+\frac {6}{5}\right ) \operatorname {hypergeom}\left (\left [\sqrt {2}-1, 2+\sqrt {2}\right ], \left [1+2 \sqrt {2}\right ], -x \right ) \left (\int \frac {x^{-1-\sqrt {2}} \left (x^{2}+1\right ) \left (\left (\left (-2 x^{2}-4 x -\frac {5}{2}\right ) \sqrt {2}+x^{2}+\frac {11 x}{2}+3\right ) \operatorname {hypergeom}\left (\left [-1-\sqrt {2}, -1-\sqrt {2}\right ], \left [1-2 \sqrt {2}\right ], -x \right )+x \operatorname {hypergeom}\left (\left [-\sqrt {2}, -\sqrt {2}\right ], \left [2-2 \sqrt {2}\right ], -x \right ) \left (x^{2}+\frac {1}{2} x +\frac {1}{2}\right ) \left (1+\sqrt {2}\right )\right )}{\left (-7 \sqrt {2}\, \operatorname {hypergeom}\left (\left [\sqrt {2}-1, \sqrt {2}-1\right ], \left [1+2 \sqrt {2}\right ], -x \right )+4 x \operatorname {hypergeom}\left (\left [\sqrt {2}, \sqrt {2}\right ], \left [2+2 \sqrt {2}\right ], -x \right ) \left (\sqrt {2}-\frac {11}{8}\right )\right ) \operatorname {hypergeom}\left (\left [-1-\sqrt {2}, -1-\sqrt {2}\right ], \left [1-2 \sqrt {2}\right ], -x \right )+4 x \operatorname {hypergeom}\left (\left [-\sqrt {2}, -\sqrt {2}\right ], \left [2-2 \sqrt {2}\right ], -x \right ) \operatorname {hypergeom}\left (\left [\sqrt {2}-1, \sqrt {2}-1\right ], \left [1+2 \sqrt {2}\right ], -x \right ) \left (\sqrt {2}+\frac {11}{8}\right )}d x \right )+2 x^{-\sqrt {2}} \operatorname {hypergeom}\left (\left [2-\sqrt {2}, -1-\sqrt {2}\right ], \left [1-2 \sqrt {2}\right ], -x \right ) c_{2} +2 x^{\sqrt {2}} \operatorname {hypergeom}\left (\left [\sqrt {2}-1, 2+\sqrt {2}\right ], \left [1+2 \sqrt {2}\right ], -x \right ) c_{1}}{2 x} \]
✓ Solution by Mathematica
Time used: 7.882 (sec). Leaf size: 636
DSolve[x^2*(x+1)*y''[x]+x*(4*x+3)*y'[x]-y[x]==x+1/x,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to x^{-1-\sqrt {2}} \left (x^{2 \sqrt {2}} \operatorname {Hypergeometric2F1}\left (-1+\sqrt {2},2+\sqrt {2},1+2 \sqrt {2},-x\right ) \int _1^x\frac {7 \operatorname {Hypergeometric2F1}\left (-1-\sqrt {2},2-\sqrt {2},1-2 \sqrt {2},-K[2]\right ) K[2]^{-1-\sqrt {2}} \left (K[2]^2+1\right )}{(K[2]+1) \left (\left (4+\sqrt {2}\right ) \operatorname {Hypergeometric2F1}\left (-\sqrt {2},3-\sqrt {2},2-2 \sqrt {2},-K[2]\right ) \operatorname {Hypergeometric2F1}\left (-1+\sqrt {2},2+\sqrt {2},1+2 \sqrt {2},-K[2]\right ) K[2]+\operatorname {Hypergeometric2F1}\left (-1-\sqrt {2},2-\sqrt {2},1-2 \sqrt {2},-K[2]\right ) \left (14 \sqrt {2} \operatorname {Hypergeometric2F1}\left (-1+\sqrt {2},2+\sqrt {2},1+2 \sqrt {2},-K[2]\right )+\left (-4+\sqrt {2}\right ) \operatorname {Hypergeometric2F1}\left (\sqrt {2},3+\sqrt {2},2 \left (1+\sqrt {2}\right ),-K[2]\right ) K[2]\right )\right )}dK[2]+\operatorname {Hypergeometric2F1}\left (-1-\sqrt {2},2-\sqrt {2},1-2 \sqrt {2},-x\right ) \int _1^x-\frac {7 \operatorname {Hypergeometric2F1}\left (-1+\sqrt {2},2+\sqrt {2},1+2 \sqrt {2},-K[1]\right ) K[1]^{-1+\sqrt {2}} \left (K[1]^2+1\right )}{(K[1]+1) \left (\left (4+\sqrt {2}\right ) \operatorname {Hypergeometric2F1}\left (-\sqrt {2},3-\sqrt {2},2-2 \sqrt {2},-K[1]\right ) \operatorname {Hypergeometric2F1}\left (-1+\sqrt {2},2+\sqrt {2},1+2 \sqrt {2},-K[1]\right ) K[1]+\operatorname {Hypergeometric2F1}\left (-1-\sqrt {2},2-\sqrt {2},1-2 \sqrt {2},-K[1]\right ) \left (14 \sqrt {2} \operatorname {Hypergeometric2F1}\left (-1+\sqrt {2},2+\sqrt {2},1+2 \sqrt {2},-K[1]\right )+\left (-4+\sqrt {2}\right ) \operatorname {Hypergeometric2F1}\left (\sqrt {2},3+\sqrt {2},2 \left (1+\sqrt {2}\right ),-K[1]\right ) K[1]\right )\right )}dK[1]+c_2 x^{2 \sqrt {2}} \operatorname {Hypergeometric2F1}\left (-1+\sqrt {2},2+\sqrt {2},1+2 \sqrt {2},-x\right )+c_1 \operatorname {Hypergeometric2F1}\left (-1-\sqrt {2},2-\sqrt {2},1-2 \sqrt {2},-x\right )\right ) \]