2.2.12 Problem 12

Maple
Mathematica
Sympy

Internal problem ID [8816]
Book : Own collection of miscellaneous problems
Section : section 2.0
Problem number : 12
Date solved : Thursday, March 13, 2025 at 06:09:46 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

Solve

yaxybxycx2=0

Maple. Time used: 0.024 (sec). Leaf size: 95
ode:=diff(diff(y(x),x),x)-a*x*diff(y(x),x)-b*x*y(x)-c*x^2 = 0; 
dsolve(ode,y(x), singsol=all);
 
y=ebxaKummerM(b22a3,12,(a2x+2b)22a3)c2b2+ebxaKummerU(b22a3,12,(a2x+2b)22a3)c1b2+c(bx+a)b2

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`
 

Mathematica. Time used: 2.789 (sec). Leaf size: 569
ode=D[y[x],{x,2}]-a*x*D[y[x],x]-b*x*y[x]-c*x^2==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 
y(x)ebxa(HermiteH(b2a3,xa2+2b2a3/2)1xa4cebK[1]aHypergeometric1F1(b22a3,12,(K[1]a2+2b)22a3)K[1]2b2(2HermiteH(b2a31,K[1]a2+2b2a3/2)Hypergeometric1F1(b22a3,12,(K[1]a2+2b)22a3)a3/2+HermiteH(b2a3,K[1]a2+2b2a3/2)Hypergeometric1F1(1b22a3,32,(K[1]a2+2b)22a3)(K[1]a2+2b))dK[1]+Hypergeometric1F1(b22a3,12,(xa2+2b)22a3)1xa4cebK[2]aHermiteH(b2a3,K[2]a2+2b2a3/2)K[2]2b2(2HermiteH(b2a31,K[2]a2+2b2a3/2)Hypergeometric1F1(b22a3,12,(K[2]a2+2b)22a3)a3/2+HermiteH(b2a3,K[2]a2+2b2a3/2)Hypergeometric1F1(1b22a3,32,(K[2]a2+2b)22a3)(K[2]a2+2b))dK[2]+c2Hypergeometric1F1(b22a3,12,(xa2+2b)22a3)+c1HermiteH(b2a3,xa2+2b2a3/2))
Sympy
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**2 + 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*x) + Derivative(y(x), (x, 2)))/(a*x) cannot be solved by the factorable group method