Internal problem ID [15521]
Internal file name [OUTPUT/15522_Friday_May_10_2024_05_47_33_PM_68571445/index.tex]
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.
ODE order: 1.
ODE degree: 1.
The type(s) of ODE detected by this program : "system of linear ODEs" Unable to solve or complete the solution.
Solve \begin {align*} x^{\prime }\left (t \right )&=\sin \left (x \left (t \right )\right ) \cos \left (y \left (t \right )\right )\\ y^{\prime }\left (t \right )&=\cos \left (x \left (t \right )\right ) \sin \left (y \left (t \right )\right ) \end {align*}
Does not currently support non linear system of equations. This is the phase plot of the system.
✓ Solution by Maple
Time used: 0.172 (sec). Leaf size: 38
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 ) &= \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: 0.492 (sec). Leaf size: 121
DSolve[{x'[t]==Sin[x[t]]*Cos[y[t]],y'[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*}