Internal
problem
ID
[8882]
Book
:
Own
collection
of
miscellaneous
problems
Section
:
section
3.0
Problem
number
:
24
Date
solved
:
Friday, February 21, 2025 at 08:49:52 PM
CAS
classification
:
[NONE]
Solve
`Methods for second order ODEs: --- Trying classification methods --- trying 2nd order Liouville trying 2nd order WeierstrassP trying 2nd order JacobiSN differential order: 2; trying a linearization to 3rd order trying 2nd order ODE linearizable_by_differentiation trying 2nd order, 2 integrating factors of the form mu(x,y) trying differential order: 2; missing variables -> trying 2nd order, dynamical_symmetries, fully reducible to Abel through one integrating factor of the form G(x,y)/(1+H(x,y)*y)^2 --- trying a change of variables {x -> y(x), y(x) -> x} and re-entering methods for dynamical symmetries --- -> trying 2nd order, dynamical_symmetries, fully reducible to Abel through one integrating factor of the form G(x,y)/(1+H(x,y)*y) trying 2nd order, integrating factors of the form mu(x,y)/(y)^n, only the singular cases trying symmetries linear in x and y(x) trying differential order: 2; exact nonlinear trying 2nd order, integrating factor of the form mu(y) trying 2nd order, integrating factor of the form mu(x,y) -> Calling odsolve with the ODE`, -(_y1^2*x^2+_y1^2-4*x^2)*y(x)/((x^2+1)*_y1^2)+(2*x^3*(diff(y(x), x))+_y1*x^2+2*(diff(y(x), x))*x-_ Methods for first order ODEs: --- Trying classification methods --- trying a quadrature trying 1st order linear <- 1st order linear successful trying 2nd order, integrating factor of the form mu(x,y)/(y)^n, only the general case trying 2nd order, integrating factor of the form mu(y,y) trying differential order: 2; mu polynomial in y trying 2nd order, integrating factor of the form mu(x,y) differential order: 2; looking for linear symmetries -> trying 2nd order, the S-function method -> trying a change of variables {x -> y(x), y(x) -> x} and re-entering methods for the S-function -> trying 2nd order, the S-function method -> trying 2nd order, No Point Symmetries Class V --- trying a change of variables {x -> y(x), y(x) -> x} and re-entering methods for dynamical symmetries --- -> trying 2nd order, No Point Symmetries Class V -> trying 2nd order, No Point Symmetries Class V --- trying a change of variables {x -> y(x), y(x) -> x} and re-entering methods for dynamical symmetries --- -> trying 2nd order, No Point Symmetries Class V -> trying 2nd order, No Point Symmetries Class V --- trying a change of variables {x -> y(x), y(x) -> x} and re-entering methods for dynamical symmetries --- -> trying 2nd order, No Point Symmetries Class V trying 2nd order, integrating factor of the form mu(x,y)/(y)^n, only the general case -> trying 2nd order, dynamical_symmetries, only a reduction of order through one integrating factor of the form G(x,y)/(1+H(x,y)*y)^ --- trying a change of variables {x -> y(x), y(x) -> x} and re-entering methods for dynamical symmetries --- -> trying 2nd order, dynamical_symmetries, only a reduction of order through one integrating factor of the form G(x,y)/(1+H(x,y)* --- Trying Lie symmetry methods, 2nd order --- `, `-> Computing symmetries using: way = 3 `, `-> Computing symmetries using: way = 5 `, `-> Computing symmetries using: way = formal`
Solving time : 0.115
(sec)
Leaf size : maple_leaf_size
dsolve((x^2+1)*diff(diff(y(x),x),x)+y(x)*diff(y(x),x)^2 = 0,y(x),singsol=all)
Solving time : 0.0
(sec)
Leaf size : 0
DSolve[{(1+x^2)*D[y[x],{x,2}]+y[x]*(D[y[x],x])^2==0,{}},y[x],x,IncludeSingularSolutions->True]
Not solved
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") y = Function("y") ode = Eq((x**2 + 1)*Derivative(y(x), (x, 2)) + y(x)*Derivative(y(x), x)**2,0) ics = {} dsolve(ode,func=y(x),ics=ics)
NotImplementedError : The given ODE -sqrt(-(x**2 + 1)*Derivative(y(x), (x, 2))/y(x)) + Derivative(y(x), x) cannot be solved by the factorable group method