60.7.164 problem 1782 (book 6.191)

Internal problem ID [11714]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 6, non-linear second order
Problem number : 1782 (book 6.191)
Date solved : Wednesday, March 05, 2025 at 02:42:29 PM
CAS classification : [[_2nd_order, _missing_x], _Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]]

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

✓ Maple. Time used: 0.144 (sec). Leaf size: 19

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

\begin{align*} y &= -i \\ y &= i \\ y &= \tan \left (\ln \left (c_{1} x +c_{2} \right )\right ) \\ \end{align*}

✓ Mathematica. Time used: 0.409 (sec). Leaf size: 153

ode=(1 - 2*y[x])*D[y[x],x]^2 + (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 {$\#$1}}\frac {\exp \left (-\int _1^{K[2]}\frac {2 K[1]-1}{K[1]^2+1}dK[1]\right )}{c_1}dK[2]\&\right ][x+c_2] \\ y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}-\frac {\exp \left (-\int _1^{K[2]}\frac {2 K[1]-1}{K[1]^2+1}dK[1]\right )}{c_1}dK[2]\&\right ][x+c_2] \\ y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {\exp \left (-\int _1^{K[2]}\frac {2 K[1]-1}{K[1]^2+1}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 - 2*y(x))*Derivative(y(x), x)**2 + (y(x)**2 + 1)*Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

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