82.6.2 problem 33-1 (b)

Internal problem ID [21849]
Book : The Differential Equations Problem Solver. VOL. II. M. Fogiel director. REA, NY. 1978. ISBN 78-63609
Section : Chapter 33. Systems of ordinary differential equations. Page 1059
Problem number : 33-1 (b)
Date solved : Sunday, October 12, 2025 at 05:51:36 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}y \left (t \right )&=-\sqrt {1-y \left (t \right )^{2}}\\ \frac {d}{d t}x \left (t \right )&=x \left (t \right )+2 y \left (t \right ) \end{align*}

✓ Maple. Time used: 0.137 (sec). Leaf size: 88

ode:=[diff(y(t),t) = -(1-y(t)^2)^(1/2), diff(x(t),t) = x(t)+2*y(t)]; 
dsolve(ode);
 

\begin{align*} [\{y \left (t \right ) = -1\}, \{x \left (t \right ) &= 2+{\mathrm e}^{t} c_1\}] \\ [\{y \left (t \right ) = 1\}, \{x \left (t \right ) &= -2+{\mathrm e}^{t} c_1\}] \\ \left [\{y \left (t \right ) &= -\sin \left (-t +c_2 \right ), y \left (t \right ) = \sin \left (-t +c_2 \right )\}, \left \{x \left (t \right ) = \left (\int -\frac {2 y \left (t \right ) \left (\frac {d}{d t}y \left (t \right )\right ) {\mathrm e}^{-t}}{\sqrt {1-y \left (t \right )^{2}}}d t +c_1 \right ) {\mathrm e}^{t}\right \}\right ] \\ \end{align*}

✓ Mathematica. Time used: 0.058 (sec). Leaf size: 38

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

\begin{align*} y(t)&\to -\sin (t-c_1)\\ x(t)&\to c_2 e^t+\cos (t-c_1)+\sin (t-c_1) \end{align*}

✓ Sympy. Time used: 0.300 (sec). Leaf size: 37

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

\[ \left [ y{\left (t \right )} = - \sin {\left (C_{2} + t \right )}, \ x{\left (t \right )} = C_{1} e^{\operatorname {asin}{\left (\sin {\left (C_{2} + t \right )} \right )}} + \sqrt {1 - \sin ^{2}{\left (C_{2} + t \right )}} + \sin {\left (C_{2} + t \right )}\right ] \]