75.28.2 problem 788

Internal problem ID [17178]
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 : Monday, March 31, 2025 at 03:43:57 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*}

✓ Maple. Time used: 0.182 (sec). Leaf size: 23

ode:=[diff(x(t),t) = -1/y(t), diff(y(t),t) = 1/x(t)]; 
dsolve(ode);
 

\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*}

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

ode={D[x[t],t]==-1/y[t],D[y[t],t]==1/x[t]}; 
ic={}; 
DSolve[{ode,ic},{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*}

✓ Sympy. Time used: 0.268 (sec). Leaf size: 22

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
ode=[Eq(Derivative(x(t), t) + 1/y(t),0),Eq(Derivative(y(t), t) - 1/x(t),0)] 
ics = {} 
dsolve(ode,func=[x(t),y(t)],ics=ics)
 

\[ \left [ x{\left (t \right )} = e^{\frac {- C_{2} - t}{C_{1}}}, \ y{\left (t \right )} = C_{1} e^{- \frac {- C_{2} - t}{C_{1}}}\right ] \]