75.28.5 problem 791

Internal problem ID [17252]
Book : A book of problems in ordinary differential equations. M.L. KRASNOV, A.L. KISELYOV, G.I. MARKARENKO. MIR, MOSCOW. 1983
Section : Chapter 3 (Systems of differential equations). Section 21. Finding integrable combinations. Exercises page 219
Problem number : 791
Date solved : Tuesday, January 28, 2025 at 08:27:39 PM
CAS classification : system_of_ODEs

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

✓ Solution by Maple

Time used: 1.702 (sec). Leaf size: 39

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

\begin{align*} \left \{y \left (t \right ) &= \pi -\operatorname {arccot}\left (\frac {\left (c_{1} {\mathrm e}^{-2 t}-c_{2} \right ) {\mathrm e}^{t}}{2}\right )\right \} \\ \left \{x \left (t \right ) &= \arccos \left (\frac {\frac {d}{d t}y \left (t \right )}{\sin \left (y \left (t \right )\right )}\right )\right \} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.437 (sec). Leaf size: 121

DSolve[{D[x[t],t]==Sin[x[t]]*Cos[y[t]],D[y[t],t]==Cos[x[t]]*Sin[y[t]]},{x[t],y[t]},t,IncludeSingularSolutions -> True]
 

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