Internal
problem
ID
[13920]
Book
:
Handbook
of
exact
solutions
for
ordinary
differential
equations.
By
Polyanin
and
Zaitsev.
Second
edition
Section
:
Chapter
2,
Second-Order
Differential
Equations.
section
2.1.2-8.
Other
equations.
Problem
number
:
260
Date
solved
:
Friday, December 19, 2025 at 08:37:56 PM
CAS
classification
:
[[_2nd_order, _with_linear_symmetries]]
ode:=(a*x^(n+1)+b*x^n+c)^2*diff(diff(y(x),x),x)+(alpha*x^n+beta*x^(n-1)+gamma)*diff(y(x),x)+(n*(-a*n-a+alpha)*x^(n-1)+(n-1)*(-b*n+beta)*x^(-2+n))*y(x) = 0; dsolve(ode,y(x), singsol=all);
Maple trace
Methods for second order ODEs: --- Trying classification methods --- trying a symmetry of the form [xi=0, eta=F(x)] checking if the LODE is missing y -> Trying an equivalence, under non-integer power transformations, to LODEs admitting Liouvillian solutions. -> Trying a solution in terms of special functions: -> Bessel -> elliptic -> Legendre -> Whittaker -> 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 @ Moebi\ us -> Heun: Equivalence to the GHE or one of its 4 confluent cases under a power \ @ Moebius -> 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 changes of variables to rationalize or make the ODE simpler <- unable to find a useful change of variables trying a symmetry of the form [xi=0, eta=F(x)] trying differential order: 2; exact nonlinear 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 symmetry of the form [xi=0, eta=F(x)] checking if the LODE is missing y -> Trying an equivalence, under non-integer power transformations, to LODEs admitting Liouvillian solutions. -> Trying a solution in terms of special functions: -> Bessel -> elliptic -> Legendre -> Whittaker -> 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 -> Heun: Equivalence to the GHE or one of its 4 confluent cases under a pow\ er @ Moebius -> 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 changes of variables to rationalize or make the ODE simpler <- unable to find a useful change of variables trying a symmetry of the form [xi=0, eta=F(x)] trying to convert to an ODE of Bessel type -> trying reduction of order to Riccati trying Riccati sub-methods: trying Riccati_symmetries -> 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] <- successful computation of symmetries. -> Computing symmetries using: way = 5 [0, y], [0, y*(a^2*(x^n)^2*x^2+2*a*(x^n)^2*x*b+2*a*x^n*x*c+b^2*(x^n)^2+2*b*x^n* c+c^2)/(a*x^(n+1)+b*x^n+c)^2] <- successful computation of symmetries. Try integration with the canonical coordinates of the symmetry [0, y*(a^2*x^(2*\ n+2)+2*a*b*x^(2*n+1)+2*x^(n+1)*a*c+b^2*x^(2*n)+2*b*x^n*c+c^2)/(a*x^(n+1)+b*x^n+\ c)^2] -> Calling odsolve with the ODE, diff(diff(y(x),x),x) = -((alpha*x^n+beta*x^(n-\ 1)+gamma)*diff(y(x),x)+(n*(-a*n-a+alpha)*x^(n-1)+(n-1)*(-b*n+beta)*x^(n-2))*y(x ))/(a*x^(n+1)+b*x^n+c)^2, y(x) *** Sublevel 2 *** Methods for second order ODEs: --- Trying classification methods --- trying a symmetry of the form [xi=0, eta=F(x)] checking if the LODE is missing y -> Trying an equivalence, under non-integer power transformations, to LODEs admitting Liouvillian solutions. -> Trying a solution in terms of special functions: -> Bessel -> elliptic -> Legendre -> Whittaker -> 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 -> Heun: Equivalence to the GHE or one of its 4 confluent cases under a pow\ er @ Moebius -> 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 changes of variables to rationalize or make the ODE simpler <- unable to find a useful change of variables trying a symmetry of the form [xi=0, eta=F(x)] trying 2nd order exact linear 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 -> Calling odsolve with the ODE, diff(_b(_a),_a) = -(2*_a^(n+1)*_b(_a)^2*a*c+2* _a^n*_b(_a)^2*b*c+_a^(2*n)*_b(_a)^2*b^2+_a^(2*n+2)*_b(_a)^2*a^2+2*a*_a^(2*n+1)* b*_b(_a)^2+_b(_a)^2*c^2-_a^(n-1)*a*n^2-_a^(n-2)*b*n^2+_a^n*_b(_a)*alpha+_a^(n-1 )*_b(_a)*beta-_a^(n-1)*a*n+_a^(n-1)*alpha*n+_a^(n-2)*b*n+_a^(n-2)*beta*n+_b(_a) *gamma-_a^(n-2)*beta)/(a*_a^(n+1)+b*_a^n+c)^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 Riccati sub-methods: trying Riccati_symmetries trying inverse_Riccati trying 1st order ODE linearizable_by_differentiation -> Computing symmetries using: way = 3 [0, y] <- successful computation of symmetries. -> Computing symmetries using: way = 5 [0, y], [0, y*(a^2*(x^n)^2*x^2+2*a*(x^n)^2*x*b+2*a*x^n*x*c+b^2*(x^n)^2+2*b*x^n* c+c^2)/(a*x^(n+1)+b*x^n+c)^2] <- successful computation of symmetries.
ode=(a*x^(n+1)+b*x^n+c)^2*D[y[x],{x,2}]+(\[Alpha]*x^n+\[Beta]*x^(n-1)+\[Gamma])*D[y[x],x]+(n*(\[Alpha]-a-a*n)*x^(n-1)+(n-1)*(\[Beta]-b*n)*x^(n-2))*y[x]==0; ic={}; DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
Not solved
from sympy import * x = symbols("x") Alpha = symbols("Alpha") BETA = symbols("BETA") Gamma = symbols("Gamma") a = symbols("a") b = symbols("b") c = symbols("c") n = symbols("n") y = Function("y") ode = Eq((n*x**(n - 1)*(Alpha - a*n - a) + x**(n - 2)*(BETA - b*n)*(n - 1))*y(x) + (Alpha*x**n + BETA*x**(n - 1) + Gamma)*Derivative(y(x), x) + (a*x**(n + 1) + b*x**n + c)**2*Derivative(y(x), (x, 2)),0) ics = {} dsolve(ode,func=y(x),ics=ics)
TypeError : Add object cannot be interpreted as an integer