54.10.1 problem 1913

Internal problem ID [13135]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 9, system of higher order odes
Problem number : 1913
Date solved : Sunday, October 12, 2025 at 02:35:46 AM
CAS classification : system_of_ODEs

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

✓ Maple. Time used: 0.220 (sec). Leaf size: 54

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

\begin{align*} \left [\{x \left (t \right ) = 0\}, \left \{y \left (t \right ) &= \frac {1}{-t +c_1}\right \}\right ] \\ \left [\left \{x \left (t \right ) &= \frac {\tanh \left (\frac {c_2 +t}{c_1}\right )}{c_1}\right \}, \left \{y \left (t \right ) = -\frac {x \left (t \right )^{2}+\frac {d}{d t}x \left (t \right )}{x \left (t \right )}\right \}\right ] \\ \end{align*}

✓ Mathematica. Time used: 0.011 (sec). Leaf size: 63

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

\begin{align*} y(t)&\to \frac {c_1}{\text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {1}{K[1]^2+c_1}dK[1]\&\right ][-t+c_2]}\\ x(t)&\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {1}{K[1]^2+c_1}dK[1]\&\right ][-t+c_2] \end{align*}

✗ Sympy

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

NotImplementedError : multiple generators [log(-C1*sqrt(-1/C1) + v), log(C1*sqrt(-1/C1) + v)] 
No algorithms are implemented to solve equation -C2 - t - sqrt(-1/C1)*log(-C1*sqrt(-1/C1) + v)/2 + sqrt(-1/C1)*log(C1*sqrt(-1/C1) + v)/2