Internal
problem
ID
[8846]
Book
:
Own
collection
of
miscellaneous
problems
Section
:
section
2.0
Problem
number
:
42
Date
solved
:
Friday, April 25, 2025 at 05:13:58 PM
CAS
classification
:
[[_2nd_order, _with_linear_symmetries]]
ode:=diff(diff(y(x),x),x)-x^3*diff(y(x),x)-x^3*y(x)-x^3 = 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 -> hyper3: Equivalence to 2F1, 1F1 or 0F1 under a power @ Moebius -> Mathieu -> Equivalence to the rational form of Mathieu ODE under a power @ Mo\ ebius trying a solution in terms of MeijerG functions -> Heun: Equivalence to the GHE or one of its 4 confluent cases under a pow\ er @ Moebius -> trying a solution of the form r0(x) * Y + r1(x) * Y where Y = exp(int(r(\ x), dx)) * 2F1([a1, a2], [b1], f) trying a symmetry of the form [xi=0, eta=F(x)] trying symmetries linear in x and y(x) trying to convert to a linear ODE with constant coefficients trying 2nd order, integrating factor of the form mu(x,y) -> Trying a solution in terms of special functions: -> Bessel -> elliptic -> Legendre -> Kummer -> hyper3: Equivalence to 1F1 under a power @ Moebius -> hypergeometric -> heuristic approach -> hyper3: Equivalence to 2F1, 1F1 or 0F1 under a power @ Moebius -> Mathieu -> Equivalence to the rational form of Mathieu ODE under a power @\ Moebius trying 2nd order exact linear trying symmetries linear in x and y(x) trying to convert to a linear ODE with constant coefficients trying to convert to an ODE of Bessel type trying to convert to an ODE of Bessel type -> trying reduction of order to Riccati trying Riccati sub-methods: -> trying a symmetry pattern of the form [F(x)*G(y), 0] -> trying a symmetry pattern of the form [0, F(x)*G(y)] -> trying a symmetry pattern of the form [F(x),G(x)*y+H(x)] --- Trying Lie symmetry methods, 2nd order --- -> Computing symmetries using: way = 3 [0, y+1] <- successful computation of symmetries. -> Computing symmetries using: way = 5 Try integration with the canonical coordinates of the symmetry [0, y+1] -> Calling odsolve with the ODE, diff(diff(y(x),x),x) = x^3*diff(y(x),x)+x^3*y( x)+x^3, y(x) *** Sublevel 2 *** 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 -> hyper3: Equivalence to 2F1, 1F1 or 0F1 under a power @ Moebius -> Mathieu -> Equivalence to the rational form of Mathieu ODE under a power @\ Moebius trying a solution in terms of MeijerG functions -> Heun: Equivalence to the GHE or one of its 4 confluent cases under a \ power @ Moebius -> trying a solution of the form r0(x) * Y + r1(x) * Y where Y = exp(int\ (r(x), dx)) * 2F1([a1, a2], [b1], f) trying a symmetry of the form [xi=0, eta=F(x)] trying symmetries linear in x and y(x) trying to convert to a linear ODE with constant coefficients trying 2nd order, integrating factor of the form mu(x,y) trying to convert to an ODE of Bessel type -> trying reduction of order to Riccati -> Calling odsolve with the ODE, diff(_b(_a),_a) = _a^3*_b(_a)+_a^3-_b(_a)^2, _b(_a), explicit *** Sublevel 2 *** Methods for first order ODEs: --- Trying classification methods --- trying a quadrature trying 1st order linear trying Bernoulli trying separable trying inverse linear trying homogeneous types: trying Chini differential order: 1; looking for linear symmetries trying exact Looking for potential symmetries trying Riccati trying inverse_Riccati trying 1st order ODE linearizable_by_differentiation -> Computing symmetries using: way = 3 [0, y+1] <- successful computation of symmetries. -> Computing symmetries using: way = 5
ode=D[y[x],{x,2}]-x^3*D[y[x],x]-x^3*y[x]-x^3==0; ic={}; DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
Not solved
from sympy import * x = symbols("x") y = Function("y") ode = Eq(-x**3*y(x) - x**3*Derivative(y(x), x) - x**3 + Derivative(y(x), (x, 2)),0) ics = {} dsolve(ode,func=y(x),ics=ics)
NotImplementedError : The given ODE y(x) + Derivative(y(x), x) + 1 - Derivative(y(x), (x, 2))/x**3 cannot be solved by the factorable group method