75.26.6 problem 773

Internal problem ID [17155]
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 : 773
Date solved : Friday, March 14, 2025 at 04:49:23 AM
CAS classification : system_of_ODEs

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

✓ Maple. Time used: 0.188 (sec). Leaf size: 61

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

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

✓ Mathematica. Time used: 63.812 (sec). Leaf size: 45

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

\begin{align*} x(t)\to \frac {t^2+c_1 t+c_2}{t+c_1} \\ y(t)\to \frac {c_1 t+c_1{}^2-c_2}{t+c_1} \\ \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 :