75.28.3 problem 789

Internal problem ID [17171]
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 : Friday, March 14, 2025 at 04:49:28 AM
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*}

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

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

\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*}

✓ Mathematica. Time used: 0.06 (sec). Leaf size: 45

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

✓ Sympy. Time used: 0.304 (sec). Leaf size: 56

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

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