54.1.59 problem 60

Internal problem ID [11373]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, linear first order
Problem number : 60
Date solved : Tuesday, September 30, 2025 at 08:07:18 PM
CAS classification : [_separable]

\begin{align*} y^{\prime }-\frac {\sqrt {y^{2}-1}}{\sqrt {x^{2}-1}}&=0 \end{align*}

✓ Maple. Time used: 0.001 (sec). Leaf size: 29

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

\[ \ln \left (x +\sqrt {x^{2}-1}\right )-\ln \left (y+\sqrt {y^{2}-1}\right )+c_1 = 0 \]

✓ Mathematica. Time used: 0.195 (sec). Leaf size: 54

ode=D[y[x],x] - Sqrt[y[x]^2-1]/Sqrt[x^2-1]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

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

✓ Sympy. Time used: 0.557 (sec). Leaf size: 17

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

\[ y{\left (x \right )} = \cosh {\left (C_{1} + \log {\left (x + \sqrt {x^{2} - 1} \right )} \right )} \]