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 : Tuesday, January 28, 2025 at 02:38:53 PM
CAS classification : system_of_ODEs

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

✓ Solution by Maple

Time used: 0.555 (sec). Leaf size: 53

dsolve([diff(x(t),t)=cos(y(t)),diff(y(t),t)=sin(x(t))-1],singsol=all)
 

\begin{align*} \left \{y &= \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 (y^{\prime }+1\right )\} \\ \end{align*}

✓ Solution by Mathematica

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

DSolve[{D[x[t],t]==Cos[y[t]],D[y[t],t]==Sin[x[t]]-1},{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*}