28.2 problem 788

Internal problem ID [15519]

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.
ODE order: 1.
ODE degree: 1.

Solve \begin {align*} x^{\prime }\left (t \right )&=-\frac {1}{y \left (t \right )}\\ y^{\prime }\left (t \right )&=\frac {1}{x \left (t \right )} \end {align*}

✓ Solution by Maple

Time used: 0.063 (sec). Leaf size: 24

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: 0.012 (sec). Leaf size: 35

DSolve[{x'[t]==-1/y[t],y'[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*}