14.28.2 problem 6

Internal problem ID [2794]
Book : Differential equations and their applications, 4th ed., M. Braun
Section : Chapter 4. Qualitative theory of differential equations. Section 4.1 (Introduction). Page 377
Problem number : 6
Date solved : Sunday, March 30, 2025 at 12:20:37 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )&=\cos \left (y \left (t \right )\right )\\ \frac {d}{d t}y \left (t \right )&=\sin \left (x \left (t \right )\right )-1 \end{align*}

✓ Maple. Time used: 0.769 (sec). Leaf size: 53

ode:=[diff(x(t),t) = cos(y(t)), diff(y(t),t) = sin(x(t))-1]; 
dsolve(ode);
 

\begin{align*} \left \{y \left (t \right ) &= \operatorname {RootOf}\left (-\int _{}^{\textit {\_Z}}\frac {1}{-1+\sin \left (\operatorname {RootOf}\left (2 \sin \left (\textit {\_f} \right )+\sqrt {2 \cos \left (2 \textit {\_Z} \right )+2}+2 \textit {\_Z} +2 c_1 \right )\right )}d \textit {\_f} +t +c_2 \right )\right \} \\ \{x \left (t \right ) &= \arcsin \left (\frac {d}{d t}y \left (t \right )+1\right )\} \\ \end{align*}

✓ Mathematica. Time used: 7.962 (sec). Leaf size: 125

ode={D[x[t],t]==Cos[y[t]],D[y[t],t]==Sin[x[t]]-1}; 
ic={}; 
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} y(t)\to \arcsin \left (-\cos \left (\text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {1}{\sqrt {1-(c_1-\cos (K[1])-K[1]){}^2}}dK[1]\&\right ][t+c_2]\right )-\text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {1}{\sqrt {1-(c_1-\cos (K[1])-K[1]){}^2}}dK[1]\&\right ][t+c_2]+c_1\right ) \\ x(t)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {1}{\sqrt {1-(c_1-\cos (K[1])-K[1]){}^2}}dK[1]\&\right ][t+c_2] \\ \end{align*}

✗ Sympy

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