Internal problem ID [7462]
Internal file name [OUTPUT/6429_Sunday_June_19_2022_05_02_08_PM_66157276/index.tex]
Book: Second order enumerated odes
Section: section 2
Problem number: 22.
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.
\[ \boxed {\left (x^{2}+1\right ) y^{\prime \prime }+\left (x +1\right ) y^{\prime }+y=4 \cos \left (\ln \left (x +1\right )\right )} \]
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 <- heuristic approach successful <- hypergeometric successful <- special function solution successful <- solving first the homogeneous part of the ODE successful`
✓ Solution by Maple
Time used: 0.031 (sec). Leaf size: 408
dsolve((1+x^2)*diff(y(x),x$2)+(1+x)*diff(y(x),x)+y(x)=4*cos(ln(1+x)),y(x), singsol=all)
\[ y \left (x \right ) = \operatorname {hypergeom}\left (\left [i, -i\right ], \left [\frac {1}{2}+\frac {i}{2}\right ], \frac {1}{2}-\frac {i x}{2}\right ) c_{2} +\left (x +i\right )^{\frac {1}{2}-\frac {i}{2}} \operatorname {hypergeom}\left (\left [\frac {1}{2}+\frac {i}{2}, \frac {1}{2}-\frac {3 i}{2}\right ], \left [\frac {3}{2}-\frac {i}{2}\right ], \frac {1}{2}-\frac {i x}{2}\right ) c_{1} +80 \left (\int \frac {\left (-i x +1\right ) \cos \left (\ln \left (x +1\right )\right ) \operatorname {hypergeom}\left (\left [\frac {1}{2}+\frac {i}{2}, \frac {1}{2}-\frac {3 i}{2}\right ], \left [\frac {3}{2}-\frac {i}{2}\right ], \frac {1}{2}-\frac {i x}{2}\right )}{\left (x^{2}+1\right ) \left (10 \operatorname {hypergeom}\left (\left [\frac {1}{2}+\frac {i}{2}, \frac {1}{2}-\frac {3 i}{2}\right ], \left [\frac {3}{2}-\frac {i}{2}\right ], \frac {1}{2}-\frac {i x}{2}\right ) \left (\left (-1-i+\left (-1+i\right ) x \right ) \operatorname {hypergeom}\left (\left [1-i, 1+i\right ], \left [\frac {3}{2}+\frac {i}{2}\right ], \frac {1}{2}-\frac {i x}{2}\right )+\left (1+i\right ) \operatorname {hypergeom}\left (\left [i, -i\right ], \left [\frac {1}{2}+\frac {i}{2}\right ], \frac {1}{2}-\frac {i x}{2}\right )\right )+\operatorname {hypergeom}\left (\left [\frac {3}{2}+\frac {i}{2}, \frac {3}{2}-\frac {3 i}{2}\right ], \left [\frac {5}{2}-\frac {i}{2}\right ], \frac {1}{2}-\frac {i x}{2}\right ) \operatorname {hypergeom}\left (\left [i, -i\right ], \left [\frac {1}{2}+\frac {i}{2}\right ], \frac {1}{2}-\frac {i x}{2}\right ) \left (1+7 i+\left (7-i\right ) x \right )\right )}d x \right ) \operatorname {hypergeom}\left (\left [i, -i\right ], \left [\frac {1}{2}+\frac {i}{2}\right ], \frac {1}{2}-\frac {i x}{2}\right )-80 \left (\int \frac {\cos \left (\ln \left (x +1\right )\right ) \left (x +i\right )^{\frac {1}{2}+\frac {i}{2}} \operatorname {hypergeom}\left (\left [i, -i\right ], \left [\frac {1}{2}+\frac {i}{2}\right ], \frac {1}{2}-\frac {i x}{2}\right )}{7 \left (\frac {10 \left (\left (1-i+\left (-1-i\right ) x \right ) \operatorname {hypergeom}\left (\left [1-i, 1+i\right ], \left [\frac {3}{2}+\frac {i}{2}\right ], \frac {1}{2}-\frac {i x}{2}\right )+\left (-1+i\right ) \operatorname {hypergeom}\left (\left [i, -i\right ], \left [\frac {1}{2}+\frac {i}{2}\right ], \frac {1}{2}-\frac {i x}{2}\right )\right ) \operatorname {hypergeom}\left (\left [\frac {1}{2}+\frac {i}{2}, \frac {1}{2}-\frac {3 i}{2}\right ], \left [\frac {3}{2}-\frac {i}{2}\right ], \frac {1}{2}-\frac {i x}{2}\right )}{7}+\left (-1+\frac {i}{7}+\left (\frac {1}{7}+i\right ) x \right ) \operatorname {hypergeom}\left (\left [\frac {3}{2}+\frac {i}{2}, \frac {3}{2}-\frac {3 i}{2}\right ], \left [\frac {5}{2}-\frac {i}{2}\right ], \frac {1}{2}-\frac {i x}{2}\right ) \operatorname {hypergeom}\left (\left [i, -i\right ], \left [\frac {1}{2}+\frac {i}{2}\right ], \frac {1}{2}-\frac {i x}{2}\right )\right ) \left (x^{2}+1\right )}d x \right ) \left (x +i\right )^{\frac {1}{2}-\frac {i}{2}} \operatorname {hypergeom}\left (\left [\frac {1}{2}+\frac {i}{2}, \frac {1}{2}-\frac {3 i}{2}\right ], \left [\frac {3}{2}-\frac {i}{2}\right ], \frac {1}{2}-\frac {i x}{2}\right ) \]
✗ Solution by Mathematica
Time used: 0.0 (sec). Leaf size: 0
DSolve[(1+x^2)*y''[x]+(1+x)*y'[x]+y[x]==4*Cos[Log[1+x]],y[x],x,IncludeSingularSolutions -> True]
Not solved