75.3.14 problem 14

Internal problem ID [19918]
Book : A short course on differential equations. By Donald Francis Campbell. Maxmillan company. London. 1907
Section : Chapter III. Ordinary differential equations of the first order and first degree. Exercises at page 33
Problem number : 14
Date solved : Thursday, October 02, 2025 at 05:01:14 PM
CAS classification : [_separable]

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

✓ Maple. Time used: 0.001 (sec). Leaf size: 37

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

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

✓ Mathematica. Time used: 0.333 (sec). Leaf size: 60

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

\begin{align*} y(x)&\to \text {InverseFunction}\left [\sqrt {\text {$\#$1}^2-1}-\arctan \left (\sqrt {\text {$\#$1}^2-1}\right )\&\right ]\left [\arctan \left (\sqrt {x^2-1}\right )-\sqrt {x^2-1}+c_1\right ]\\ y(x)&\to 0 \end{align*}

✓ Sympy. Time used: 0.750 (sec). Leaf size: 158

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

\[ \begin {cases} - i \operatorname {acosh}{\left (\frac {1}{y{\left (x \right )}} \right )} - \frac {i y{\left (x \right )}}{\sqrt {-1 + \frac {1}{y^{2}{\left (x \right )}}}} + \frac {i}{\sqrt {-1 + \frac {1}{y^{2}{\left (x \right )}}} y{\left (x \right )}} & \text {for}\: \frac {1}{\left |{y^{2}{\left (x \right )}}\right |} > 1 \\\operatorname {asin}{\left (\frac {1}{y{\left (x \right )}} \right )} + \frac {y{\left (x \right )}}{\sqrt {1 - \frac {1}{y^{2}{\left (x \right )}}}} - \frac {1}{\sqrt {1 - \frac {1}{y^{2}{\left (x \right )}}} y{\left (x \right )}} & \text {otherwise} \end {cases} = C_{1} - \begin {cases} - \frac {i x}{\sqrt {-1 + \frac {1}{x^{2}}}} - i \operatorname {acosh}{\left (\frac {1}{x} \right )} + \frac {i}{x \sqrt {-1 + \frac {1}{x^{2}}}} & \text {for}\: \frac {1}{\left |{x^{2}}\right |} > 1 \\\frac {x}{\sqrt {1 - \frac {1}{x^{2}}}} + \operatorname {asin}{\left (\frac {1}{x} \right )} - \frac {1}{x \sqrt {1 - \frac {1}{x^{2}}}} & \text {otherwise} \end {cases} \]