60.7.181 problem 1801 (book 6.210)

Internal problem ID [11731]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 6, non-linear second order
Problem number : 1801 (book 6.210)
Date solved : Sunday, March 30, 2025 at 08:44:57 PM
CAS classification : [[_2nd_order, _missing_x], _Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]]

\begin{align*} y \left (y^{2}+1\right ) y^{\prime \prime }+\left (1-3 y^{2}\right ) {y^{\prime }}^{2}&=0 \end{align*}

✓ Maple. Time used: 0.037 (sec). Leaf size: 78

ode:=y(x)*(1+y(x)^2)*diff(diff(y(x),x),x)+(1-3*y(x)^2)*diff(y(x),x)^2 = 0; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= -i \\ y &= i \\ y &= 0 \\ y &= -\frac {\sqrt {2}\, \sqrt {-2 \left (c_1 x +c_2 \right ) \left (c_1 x +c_2 +\frac {1}{2}\right )}}{2 c_1 x +2 c_2} \\ y &= \frac {\sqrt {2}\, \sqrt {-2 \left (c_1 x +c_2 \right ) \left (c_1 x +c_2 +\frac {1}{2}\right )}}{2 c_1 x +2 c_2} \\ \end{align*}

✓ Mathematica. Time used: 0.932 (sec). Leaf size: 171

ode=(1 - 3*y[x]^2)*D[y[x],x]^2 + y[x]*(1 + y[x]^2)*D[y[x],{x,2}] == 0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {\exp \left (-\int _1^{K[2]}\frac {3 K[1]^2-1}{K[1] \left (K[1]^2+1\right )}dK[1]\right )}{c_1}dK[2]\&\right ][x+c_2] \\ y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}-\frac {\exp \left (-\int _1^{K[2]}\frac {3 K[1]^2-1}{K[1] \left (K[1]^2+1\right )}dK[1]\right )}{c_1}dK[2]\&\right ][x+c_2] \\ y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {\exp \left (-\int _1^{K[2]}\frac {3 K[1]^2-1}{K[1] \left (K[1]^2+1\right )}dK[1]\right )}{c_1}dK[2]\&\right ][x+c_2] \\ \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq((1 - 3*y(x)**2)*Derivative(y(x), x)**2 + (y(x)**2 + 1)*y(x)*Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE -sqrt((y(x)**2 + 1)*y(x)*Derivative(y(x), (x, 2))/(3*y(x)**2 - 1)) + Derivative(y(x), x) cannot be solved by the factorable group method