2.2.11 Problem 11

Internal problem ID [8815]
Book : Own collection of miscellaneous problems
Section : section 2.0
Problem number : 11
Date solved : Sunday, February 23, 2025 at 04:37:35 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

Solve

\begin{align*} y^{\prime \prime }-a x y^{\prime }-b x y-c x&=0 \end{align*}

Maple step by step solution

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`
 

Maple dsolve solution

Solving time : 0.026 (sec)
Leaf size : 86

dsolve(diff(diff(y(x),x),x)-a*x*diff(y(x),x)-b*x*y(x)-c*x = 0,y(x),singsol=all)
 

\[ y = \frac {{\mathrm e}^{-\frac {b x}{a}} \operatorname {KummerM}\left (-\frac {b^{2}}{2 a^{3}}, \frac {1}{2}, \frac {\left (a^{2} x +2 b \right )^{2}}{2 a^{3}}\right ) c_{2} b +{\mathrm e}^{-\frac {b x}{a}} \operatorname {KummerU}\left (-\frac {b^{2}}{2 a^{3}}, \frac {1}{2}, \frac {\left (a^{2} x +2 b \right )^{2}}{2 a^{3}}\right ) c_{1} b -c}{b} \]

✓Mathematica DSolve solution

Solving time : 4.917 (sec)
Leaf size : 565

DSolve[{D[y[x],{x,2}]-a*x*D[y[x],x]-b*x*y[x]-c*x==0,{}},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to e^{-\frac {b x}{a}} \left (\operatorname {HermiteH}\left (\frac {b^2}{a^3},\frac {x a^2+2 b}{\sqrt {2} a^{3/2}}\right ) \int _1^x\frac {a^4 c e^{\frac {b K[1]}{a}} \operatorname {Hypergeometric1F1}\left (-\frac {b^2}{2 a^3},\frac {1}{2},\frac {\left (K[1] a^2+2 b\right )^2}{2 a^3}\right ) K[1]}{b^2 \left (\sqrt {2} \operatorname {HermiteH}\left (\frac {b^2}{a^3}-1,\frac {K[1] a^2+2 b}{\sqrt {2} a^{3/2}}\right ) \operatorname {Hypergeometric1F1}\left (-\frac {b^2}{2 a^3},\frac {1}{2},\frac {\left (K[1] a^2+2 b\right )^2}{2 a^3}\right ) a^{3/2}+\operatorname {HermiteH}\left (\frac {b^2}{a^3},\frac {K[1] a^2+2 b}{\sqrt {2} a^{3/2}}\right ) \operatorname {Hypergeometric1F1}\left (1-\frac {b^2}{2 a^3},\frac {3}{2},\frac {\left (K[1] a^2+2 b\right )^2}{2 a^3}\right ) \left (K[1] a^2+2 b\right )\right )}dK[1]+\operatorname {Hypergeometric1F1}\left (-\frac {b^2}{2 a^3},\frac {1}{2},\frac {\left (x a^2+2 b\right )^2}{2 a^3}\right ) \int _1^x-\frac {a^4 c e^{\frac {b K[2]}{a}} \operatorname {HermiteH}\left (\frac {b^2}{a^3},\frac {K[2] a^2+2 b}{\sqrt {2} a^{3/2}}\right ) K[2]}{b^2 \left (\sqrt {2} \operatorname {HermiteH}\left (\frac {b^2}{a^3}-1,\frac {K[2] a^2+2 b}{\sqrt {2} a^{3/2}}\right ) \operatorname {Hypergeometric1F1}\left (-\frac {b^2}{2 a^3},\frac {1}{2},\frac {\left (K[2] a^2+2 b\right )^2}{2 a^3}\right ) a^{3/2}+\operatorname {HermiteH}\left (\frac {b^2}{a^3},\frac {K[2] a^2+2 b}{\sqrt {2} a^{3/2}}\right ) \operatorname {Hypergeometric1F1}\left (1-\frac {b^2}{2 a^3},\frac {3}{2},\frac {\left (K[2] a^2+2 b\right )^2}{2 a^3}\right ) \left (K[2] a^2+2 b\right )\right )}dK[2]+c_2 \operatorname {Hypergeometric1F1}\left (-\frac {b^2}{2 a^3},\frac {1}{2},\frac {\left (x a^2+2 b\right )^2}{2 a^3}\right )+c_1 \operatorname {HermiteH}\left (\frac {b^2}{a^3},\frac {x a^2+2 b}{\sqrt {2} a^{3/2}}\right )\right ) \]

✗Sympy solution

Solving time : 0.000 (sec)
Leaf size : 0

Python version: 3.13.1 (main, Dec  4 2024, 18:05:56) [GCC 14.2.1 20240910] 
Sympy version 1.13.3
 

from sympy import * 
x = symbols("x") 
a = symbols("a") 
b = symbols("b") 
c = symbols("c") 
y = Function("y") 
ode = Eq(-a*x*Derivative(y(x), x) - b*x*y(x) - c*x + Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE Derivative(y(x), x) - (-x*(b*y(x) + c) + Derivative(y(x), (x, 2)))/(a*x) cannot be solved by the factorable group method