80.2.6 problem 8

Internal problem ID [21149]
Book : A Textbook on Ordinary Differential Equations by Shair Ahmad and Antonio Ambrosetti. Second edition. ISBN 978-3-319-16407-6. Springer 2015
Section : Chapter 2. Theory of first order differential equations. Excercise 2.6 at page 37
Problem number : 8
Date solved : Thursday, October 02, 2025 at 07:10:20 PM
CAS classification : [_quadrature]

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

With initial conditions

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

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

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

\[ x = 1 \]

✓ Mathematica. Time used: 0.011 (sec). Leaf size: 7

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

\begin{align*} x(t)&\to \sin (t) \end{align*}

✗ Sympy

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

NotImplementedError : Initial conditions produced too many solutions for constants