75.26.7 problem 774

Internal problem ID [17164]
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 19. Basic concepts and definitions. Exercises page 199
Problem number : 774
Date solved : Monday, March 31, 2025 at 03:43:39 PM
CAS classification : system_of_ODEs

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

✓ Maple. Time used: 0.702 (sec). Leaf size: 131

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

\begin{align*} \left \{x \left (t \right ) &= t +\operatorname {RootOf}\left (-t +\int _{}^{\textit {\_Z}}-\frac {2 \left ({\mathrm e}^{c_1} \textit {\_f}^{2}-1\right )}{-4+3 \,{\mathrm e}^{c_1} \textit {\_f}^{2}-\sqrt {-3 \,{\mathrm e}^{c_1} \textit {\_f}^{2}+4}\, {\mathrm e}^{\frac {c_1}{2}} \textit {\_f}}d \textit {\_f} +c_2 \right ), x \left (t \right ) = t +\operatorname {RootOf}\left (-t +\int _{}^{\textit {\_Z}}-\frac {2 \left ({\mathrm e}^{c_1} \textit {\_f}^{2}-1\right )}{3 \,{\mathrm e}^{c_1} \textit {\_f}^{2}+\sqrt {-3 \,{\mathrm e}^{c_1} \textit {\_f}^{2}+4}\, {\mathrm e}^{\frac {c_1}{2}} \textit {\_f} -4}d \textit {\_f} +c_2 \right )\right \} \\ \left \{y \left (t \right ) &= \frac {\left (\frac {d}{d t}x \left (t \right )\right ) x \left (t \right )+t}{\frac {d}{d t}x \left (t \right )+1}\right \} \\ \end{align*}

✓ Mathematica. Time used: 5.742 (sec). Leaf size: 151

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

\begin{align*} x(t)\to \frac {1}{2} \left (-\sqrt {-3 t^2+2 c_1 t+c_1{}^2+4 c_2}-t+c_1\right ) \\ y(t)\to \frac {1}{2} \left (\sqrt {-3 t^2+2 c_1 t+c_1{}^2+4 c_2}-t+c_1\right ) \\ x(t)\to \frac {1}{2} \left (\sqrt {-3 t^2+2 c_1 t+c_1{}^2+4 c_2}-t+c_1\right ) \\ y(t)\to \frac {1}{2} \left (-\sqrt {-3 t^2+2 c_1 t+c_1{}^2+4 c_2}-t+c_1\right ) \\ \end{align*}

✗ Sympy

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

NotImplementedError :