79.1.10 problem 2 (iv)

Internal problem ID [18418]
Book : Elementary Differential Equations. By R.L.E. Schwarzenberger. Chapman and Hall. London. First Edition (1969)
Section : Chapter 3. Solutions of first-order equations. Exercises at page 47
Problem number : 2 (iv)
Date solved : Thursday, March 13, 2025 at 11:56:01 AM
CAS classification : [_quadrature]

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

With initial conditions

\begin{align*} x \left (0\right )&=1 \end{align*}

✓ Maple. Time used: 0.003 (sec). Leaf size: 5

ode:=diff(x(t),t) = (x(t)^2-1)^(1/2); 
ic:=x(0) = 1; 
dsolve([ode,ic],x(t), singsol=all);
 

\[ x = 1 \]

✓ Mathematica. Time used: 0.009 (sec). Leaf size: 18

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

\[ x(t)\to \frac {1}{2} \left (e^{-t}+e^t\right ) \]

✓ Sympy. Time used: 0.666 (sec). Leaf size: 5

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

\[ x{\left (t \right )} = \cosh {\left (t \right )} \]