75.28.2 problem 788

Internal problem ID [17249]
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 : 788
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 {1}{y \left (t \right )}\\ \frac {d}{d t}y \left (t \right )&=\frac {1}{x \left (t \right )} \end{align*}

✓ Solution by Maple

Time used: 0.097 (sec). Leaf size: 23

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

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

✓ Solution by Mathematica

Time used: 0.015 (sec). Leaf size: 35

DSolve[{D[x[t],t]==-1/y[t],D[y[t],t]==1/x[t]},{x[t],y[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} y(t)\to \frac {c_1 e^{\frac {t}{c_1}}}{c_2} \\ x(t)\to c_2 e^{-\frac {t}{c_1}} \\ \end{align*}