75.28.4 problem 790

Internal problem ID [17180]
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 : 790
Date solved : Monday, March 31, 2025 at 03:43:59 PM
CAS classification : system_of_ODEs

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

✓ Maple. Time used: 0.126 (sec). Leaf size: 47

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

\begin{align*} \left \{x \left (t \right ) &= \frac {-c_1 \,t^{2}-2 c_2 t -2}{2 c_1 t +2 c_2}\right \} \\ \left \{y \left (t \right ) &= \frac {\left (\frac {d}{d t}x \left (t \right )\right ) x \left (t \right )}{\frac {d}{d t}x \left (t \right )+1}\right \} \\ \end{align*}

✓ Mathematica. Time used: 0.084 (sec). Leaf size: 145

ode={D[x[t],t]==y[t]/(x[t]-y[t]),D[y[t],t]==x[t]/(x[t]-y[t])}; 
ic={}; 
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} y(t)\to -\frac {1}{2} \sqrt {\frac {\left (t^2-2 c_2 t+c_2{}^2+2 c_1\right ){}^2}{(t-c_2){}^2}} \\ x(t)\to -\frac {t^2-2 c_2 t+c_2{}^2-2 c_1}{2 t-2 c_2} \\ y(t)\to \frac {1}{2} \sqrt {\frac {\left (t^2-2 c_2 t+c_2{}^2+2 c_1\right ){}^2}{(t-c_2){}^2}} \\ x(t)\to -\frac {t^2-2 c_2 t+c_2{}^2-2 c_1}{2 t-2 c_2} \\ \end{align*}

✓ Sympy. Time used: 0.544 (sec). Leaf size: 51

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

\[ \left [ x{\left (t \right )} = \frac {C_{1} - C_{2}^{2} - 2 C_{2} t - t^{2}}{2 \left (C_{2} + t\right )}, \ y{\left (t \right )} = \sqrt {C_{1} + \frac {\left (C_{1} - C_{2}^{2} - 2 C_{2} t - t^{2}\right )^{2}}{4 \left (C_{2} + t\right )^{2}}}\right ] \]