86.8.2 problem 2

Internal problem ID [23162]
Book : An introduction to Differential Equations. By Howard Frederick Cleaves. 1969. Oliver and Boyd publisher. ISBN 0050015044
Section : Chapter 5. Linear equations of the second order with constant coefficients. Exercise 5e at page 91
Problem number : 2
Date solved : Thursday, October 02, 2025 at 09:23:38 PM
CAS classification : [[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]]

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

✓ Maple. Time used: 0.023 (sec). Leaf size: 33

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

\begin{align*} y &= -1 \\ y &= 1 \\ y &= c_1 \\ \int _{}^{y}\frac {1}{\operatorname {arctanh}\left (\textit {\_a} \right )+c_1}d \textit {\_a} -x -c_2 &= 0 \\ \end{align*}

✓ Mathematica. Time used: 0.174 (sec). Leaf size: 132

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

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

✗ Sympy

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

NotImplementedError : The given ODE -(1 - y(x)**2)*Derivative(y(x), (x, 2)) + Derivative(y(x), x) ca