Internal problem ID [7149]
Internal file name [OUTPUT/6135_Sunday_June_05_2022_04_24_31_PM_59286259/index.tex]
Book: Own collection of miscellaneous problems
Section: section 2.0
Problem number: 13.
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.
\[ \boxed {y^{\prime \prime }-a x y^{\prime }-b x y=c \,x^{3}} \]
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: 0.0 (sec). Leaf size: 517
dsolve(diff(y(x),x$2)-a*x*diff(y(x),x)-b*x*y(x)-c*x^3=0,y(x), singsol=all)
\[ y \left (x \right ) = \frac {{\mathrm e}^{-\frac {b x}{a}} \left (2 \operatorname {KummerU}\left (-\frac {b^{2}}{2 a^{3}}, \frac {1}{2}, \frac {\left (a^{2} x +2 b \right )^{2}}{2 a^{3}}\right ) \left (\int -\frac {\left (a^{2} x +2 b \right ) x^{3} \operatorname {KummerM}\left (-\frac {b^{2}}{2 a^{3}}, \frac {1}{2}, \frac {\left (a^{2} x +2 b \right )^{2}}{2 a^{3}}\right ) {\mathrm e}^{\frac {b x}{a}}}{\operatorname {KummerM}\left (-\frac {b^{2}}{2 a^{3}}, \frac {1}{2}, \frac {\left (a^{2} x +2 b \right )^{2}}{2 a^{3}}\right ) \left (a^{3}-b^{2}\right ) \operatorname {KummerU}\left (\frac {2 a^{3}-b^{2}}{2 a^{3}}, \frac {1}{2}, \frac {\left (a^{2} x +2 b \right )^{2}}{2 a^{3}}\right )-2 \operatorname {KummerM}\left (\frac {2 a^{3}-b^{2}}{2 a^{3}}, \frac {1}{2}, \frac {\left (a^{2} x +2 b \right )^{2}}{2 a^{3}}\right ) \operatorname {KummerU}\left (-\frac {b^{2}}{2 a^{3}}, \frac {1}{2}, \frac {\left (a^{2} x +2 b \right )^{2}}{2 a^{3}}\right ) a^{3}}d x \right ) a^{4} c -2 \operatorname {KummerM}\left (-\frac {b^{2}}{2 a^{3}}, \frac {1}{2}, \frac {\left (a^{2} x +2 b \right )^{2}}{2 a^{3}}\right ) \left (\int -\frac {\left (a^{2} x +2 b \right ) x^{3} \operatorname {KummerU}\left (-\frac {b^{2}}{2 a^{3}}, \frac {1}{2}, \frac {\left (a^{2} x +2 b \right )^{2}}{2 a^{3}}\right ) {\mathrm e}^{\frac {b x}{a}}}{\operatorname {KummerM}\left (-\frac {b^{2}}{2 a^{3}}, \frac {1}{2}, \frac {\left (a^{2} x +2 b \right )^{2}}{2 a^{3}}\right ) \left (a^{3}-b^{2}\right ) \operatorname {KummerU}\left (\frac {2 a^{3}-b^{2}}{2 a^{3}}, \frac {1}{2}, \frac {\left (a^{2} x +2 b \right )^{2}}{2 a^{3}}\right )-2 \operatorname {KummerM}\left (\frac {2 a^{3}-b^{2}}{2 a^{3}}, \frac {1}{2}, \frac {\left (a^{2} x +2 b \right )^{2}}{2 a^{3}}\right ) \operatorname {KummerU}\left (-\frac {b^{2}}{2 a^{3}}, \frac {1}{2}, \frac {\left (a^{2} x +2 b \right )^{2}}{2 a^{3}}\right ) a^{3}}d x \right ) a^{4} c +b^{2} c_{1} \operatorname {KummerU}\left (-\frac {b^{2}}{2 a^{3}}, \frac {1}{2}, \frac {\left (a^{2} x +2 b \right )^{2}}{2 a^{3}}\right )+b^{2} c_{2} \operatorname {KummerM}\left (-\frac {b^{2}}{2 a^{3}}, \frac {1}{2}, \frac {\left (a^{2} x +2 b \right )^{2}}{2 a^{3}}\right )\right )}{b^{2}} \]
✓ Solution by Mathematica
Time used: 3.085 (sec). Leaf size: 569
DSolve[y''[x]-a*x*y'[x]-b*x*y[x]-c*x^3==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to e^{-\frac {b x}{a}} \left (\operatorname {HermiteH}\left (\frac {b^2}{a^3},\frac {x a^2+2 b}{\sqrt {2} a^{3/2}}\right ) \int _1^x\frac {a^4 c e^{\frac {b K[1]}{a}} \operatorname {Hypergeometric1F1}\left (-\frac {b^2}{2 a^3},\frac {1}{2},\frac {\left (K[1] a^2+2 b\right )^2}{2 a^3}\right ) K[1]^3}{b^2 \left (\sqrt {2} \operatorname {HermiteH}\left (\frac {b^2}{a^3}-1,\frac {K[1] a^2+2 b}{\sqrt {2} a^{3/2}}\right ) \operatorname {Hypergeometric1F1}\left (-\frac {b^2}{2 a^3},\frac {1}{2},\frac {\left (K[1] a^2+2 b\right )^2}{2 a^3}\right ) a^{3/2}+\operatorname {HermiteH}\left (\frac {b^2}{a^3},\frac {K[1] a^2+2 b}{\sqrt {2} a^{3/2}}\right ) \operatorname {Hypergeometric1F1}\left (1-\frac {b^2}{2 a^3},\frac {3}{2},\frac {\left (K[1] a^2+2 b\right )^2}{2 a^3}\right ) \left (K[1] a^2+2 b\right )\right )}dK[1]+\operatorname {Hypergeometric1F1}\left (-\frac {b^2}{2 a^3},\frac {1}{2},\frac {\left (x a^2+2 b\right )^2}{2 a^3}\right ) \int _1^x-\frac {a^4 c e^{\frac {b K[2]}{a}} \operatorname {HermiteH}\left (\frac {b^2}{a^3},\frac {K[2] a^2+2 b}{\sqrt {2} a^{3/2}}\right ) K[2]^3}{b^2 \left (\sqrt {2} \operatorname {HermiteH}\left (\frac {b^2}{a^3}-1,\frac {K[2] a^2+2 b}{\sqrt {2} a^{3/2}}\right ) \operatorname {Hypergeometric1F1}\left (-\frac {b^2}{2 a^3},\frac {1}{2},\frac {\left (K[2] a^2+2 b\right )^2}{2 a^3}\right ) a^{3/2}+\operatorname {HermiteH}\left (\frac {b^2}{a^3},\frac {K[2] a^2+2 b}{\sqrt {2} a^{3/2}}\right ) \operatorname {Hypergeometric1F1}\left (1-\frac {b^2}{2 a^3},\frac {3}{2},\frac {\left (K[2] a^2+2 b\right )^2}{2 a^3}\right ) \left (K[2] a^2+2 b\right )\right )}dK[2]+c_2 \operatorname {Hypergeometric1F1}\left (-\frac {b^2}{2 a^3},\frac {1}{2},\frac {\left (x a^2+2 b\right )^2}{2 a^3}\right )+c_1 \operatorname {HermiteH}\left (\frac {b^2}{a^3},\frac {x a^2+2 b}{\sqrt {2} a^{3/2}}\right )\right ) \]