Internal problem ID [15519]
Internal file name [OUTPUT/15520_Friday_May_10_2024_05_47_33_PM_93593442/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: 789.
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 )&=\frac {x \left (t \right )}{y \left (t \right )}\\ y^{\prime }\left (t \right )&=\frac {y \left (t \right )}{x \left (t \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: 0.125 (sec). Leaf size: 34
dsolve([diff(x(t),t)=x(t)/y(t),diff(y(t),t)=y(t)/x(t)],singsol=all)
\begin{align*} \left \{x \left (t \right ) &= \frac {-1+{\mathrm e}^{c_{2} c_{1}} {\mathrm e}^{c_{1} t}}{c_{1}}\right \} \\ \left \{y \left (t \right ) &= \frac {x \left (t \right )}{\frac {d}{d t}x \left (t \right )}\right \} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.076 (sec). Leaf size: 45
DSolve[{x'[t]==x[t]/y[t],y'[t]==y[t]/x[t]},{x[t],y[t]},t,IncludeSingularSolutions -> True]
\begin{align*} y(t)\to -\frac {e^{c_1 t}+c_1 c_2}{c_1{}^2 c_2} \\ x(t)\to c_2 e^{c_1 (-t)}+\frac {1}{c_1} \\ \end{align*}