81.3.14 problem 14
Internal
problem
ID
[18553]
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, March 13, 2025 at 12:12:51 PM
CAS
classification
:
[_separable]
\begin{align*} y \sqrt {x^{2}-1}+x \sqrt {y^{2}-1}\, y^{\prime }&=0 \end{align*}
✓ Maple. Time used: 0.001 (sec). Leaf size: 37
ode:=y(x)*(x^2-1)^(1/2)+x*(-1+y(x)^2)^(1/2)*diff(y(x),x) = 0;
dsolve(ode,y(x), singsol=all);
\[
\sqrt {x^{2}-1}+\arctan \left (\frac {1}{\sqrt {x^{2}-1}}\right )+\sqrt {-1+y \left (x \right )^{2}}+\arctan \left (\frac {1}{\sqrt {-1+y \left (x \right )^{2}}}\right )+c_{1} = 0
\]
✓ Mathematica. Time used: 0.461 (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: 1.214 (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)
\[
\begin {cases} - i \operatorname {acosh}{\left (\frac {1}{y{\left (x \right )}} \right )} - \frac {i y{\left (x \right )}}{\sqrt {-1 + \frac {1}{y^{2}{\left (x \right )}}}} + \frac {i}{\sqrt {-1 + \frac {1}{y^{2}{\left (x \right )}}} y{\left (x \right )}} & \text {for}\: \frac {1}{\left |{y^{2}{\left (x \right )}}\right |} > 1 \\\operatorname {asin}{\left (\frac {1}{y{\left (x \right )}} \right )} + \frac {y{\left (x \right )}}{\sqrt {1 - \frac {1}{y^{2}{\left (x \right )}}}} - \frac {1}{\sqrt {1 - \frac {1}{y^{2}{\left (x \right )}}} y{\left (x \right )}} & \text {otherwise} \end {cases} = C_{1} - \begin {cases} - \frac {i x}{\sqrt {-1 + \frac {1}{x^{2}}}} - i \operatorname {acosh}{\left (\frac {1}{x} \right )} + \frac {i}{x \sqrt {-1 + \frac {1}{x^{2}}}} & \text {for}\: \frac {1}{\left |{x^{2}}\right |} > 1 \\\frac {x}{\sqrt {1 - \frac {1}{x^{2}}}} + \operatorname {asin}{\left (\frac {1}{x} \right )} - \frac {1}{x \sqrt {1 - \frac {1}{x^{2}}}} & \text {otherwise} \end {cases}
\]