60.1.60 problem 61

Internal problem ID [10074]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, linear first order
Problem number : 61
Date solved : Sunday, March 30, 2025 at 03:03:22 PM
CAS classification : [_separable]

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

✓ Maple. Time used: 0.001 (sec). Leaf size: 50

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

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

✓ Mathematica. Time used: 0.688 (sec). Leaf size: 75

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

\[ y(x)\to \text {InverseFunction}\left [\frac {1}{2} \text {$\#$1} \sqrt {\text {$\#$1}^2-1}-\frac {1}{2} \log \left (\sqrt {\text {$\#$1}^2-1}+\text {$\#$1}\right )\&\right ]\left [-\frac {1}{2} \text {arctanh}\left (\frac {x}{\sqrt {x^2-1}}\right )+\frac {1}{2} \sqrt {x^2-1} x+c_1\right ] \]

✓ Sympy. Time used: 0.865 (sec). Leaf size: 58

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

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