75.28.3 problem 789

Internal problem ID [17250]
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
Date solved : Tuesday, January 28, 2025 at 08:27:38 PM
CAS classification : system_of_ODEs

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

✓ Solution by Maple

Time used: 0.949 (sec). Leaf size: 33

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: 0.060 (sec). Leaf size: 45

DSolve[{D[x[t],t]==x[t]/y[t],D[y[t],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*}