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
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);
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`
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]
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