Internal problem ID [7181]
Internal file name [OUTPUT/6167_Sunday_June_05_2022_04_26_19_PM_22928568/index.tex]
Book: Own collection of miscellaneous problems
Section: section 2.0
Problem number: 44.
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.
\[ \boxed {y^{\prime \prime }-x^{2} y^{\prime }-y x=x^{2}} \]
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 <- Bessel successful <- special function solution successful <- solving first the homogeneous part of the ODE successful`
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 44
dsolve(diff(y(x),x$2)-x^2*diff(y(x),x)-x*y(x)-x^2=0,y(x), singsol=all)
\[ y \left (x \right ) = {\mathrm e}^{\frac {x^{3}}{6}} \sqrt {x}\, \operatorname {BesselI}\left (\frac {1}{6}, \frac {x^{3}}{6}\right ) c_{2} +{\mathrm e}^{\frac {x^{3}}{6}} \sqrt {x}\, \operatorname {BesselK}\left (\frac {1}{6}, \frac {x^{3}}{6}\right ) c_{1} -\frac {x}{2} \]
✓ Solution by Mathematica
Time used: 0.344 (sec). Leaf size: 224
DSolve[y''[x]-x^2*y'[x]-x*y[x]-x^2==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to -\frac {e^{\frac {x^3}{6}} \left (12 \left (x^3\right )^{5/6} \operatorname {Gamma}\left (\frac {1}{6}\right ) \operatorname {Gamma}\left (\frac {7}{6}\right ) \operatorname {BesselI}\left (\frac {1}{6},\frac {x^3}{6}\right ) \, _1F_1\left (-\frac {2}{3};-\frac {1}{3};-\frac {x^3}{3}\right )+\sqrt [3]{2} 3^{2/3} \sqrt [6]{x^3} x^6 \operatorname {Gamma}\left (\frac {1}{6}\right ) \operatorname {Gamma}\left (\frac {5}{6}\right ) \operatorname {BesselI}\left (-\frac {1}{6},\frac {x^3}{6}\right ) \, _1F_1\left (\frac {2}{3};\frac {7}{3};-\frac {x^3}{3}\right )-4 \operatorname {Gamma}\left (\frac {7}{6}\right ) \left (6 \sqrt [3]{2} 3^{2/3} c_1 x^{5/2} \operatorname {Gamma}\left (\frac {5}{6}\right ) \operatorname {BesselI}\left (-\frac {1}{6},\frac {x^3}{6}\right )+\operatorname {Gamma}\left (\frac {1}{6}\right ) \left (3 \left (x^3\right )^{5/6}+2 \sqrt [3]{-1} 3^{2/3} c_2 x^{5/2}\right ) \operatorname {BesselI}\left (\frac {1}{6},\frac {x^3}{6}\right )\right )\right )}{24\ 2^{2/3} 3^{5/6} x^2 \operatorname {Gamma}\left (\frac {7}{6}\right )} \]