68.3.14 problem 13 (c)

Internal problem ID [17161]
Book : INTRODUCTORY DIFFERENTIAL EQUATIONS. Martha L. Abell, James P. Braselton. Fourth edition 2014. ElScAe. 2014
Section : Chapter 2. First Order Equations. Exercises 2.1, page 32
Problem number : 13 (c)
Date solved : Thursday, October 02, 2025 at 01:45:38 PM
CAS classification : [_quadrature]

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

With initial conditions

\begin{align*} y \left (0\right )&={\frac {1}{2}} \\ \end{align*}

✓ Maple. Time used: 0.214 (sec). Leaf size: 30

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

\[ y = -\frac {\left (i \sqrt {3}-1\right ) \left (2 \,{\mathrm e}^{-t}+\left (i \sqrt {3}-1\right ) {\mathrm e}^{t}\right )}{8} \]

✓ Mathematica. Time used: 0.011 (sec). Leaf size: 32

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

\begin{align*} y(t)&\to \frac {1}{2} \sqrt [3]{-1} e^{-t} \left (e^{2 t}-\sqrt [3]{-1}\right ) \end{align*}

✗ Sympy

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

NotImplementedError : Initial conditions produced too many solutions for constants