54.1.435 problem 447

Internal problem ID [11749]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, linear first order
Problem number : 447
Date solved : Tuesday, September 30, 2025 at 10:16:33 PM
CAS classification : [_quadrature]

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

✓ Maple. Time used: 0.028 (sec). Leaf size: 33

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

\begin{align*} y &= \ln \left (x +\sqrt {x^{2}-1}\right )+c_1 \\ y &= -\ln \left (x +\sqrt {x^{2}-1}\right )+c_1 \\ \end{align*}

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 41

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

\begin{align*} y(x)&\to -\log \left (\sqrt {x^2-1}+x\right )+c_1\\ y(x)&\to \log \left (\sqrt {x^2-1}+x\right )+c_1 \end{align*}

✓ Sympy. Time used: 0.646 (sec). Leaf size: 41

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

\[ \left [ y{\left (x \right )} = C_{1} - \begin {cases} \log {\left (x + \sqrt {x^{2} - 1} \right )} & \text {for}\: x > -1 \wedge x < 1 \end {cases}, \ y{\left (x \right )} = C_{1} + \begin {cases} \log {\left (x + \sqrt {x^{2} - 1} \right )} & \text {for}\: x > -1 \wedge x < 1 \end {cases}\right ] \]