54.10.15 problem 1930

Internal problem ID [13149]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 9, system of higher order odes
Problem number : 1930
Date solved : Sunday, October 12, 2025 at 02:35:49 AM
CAS classification : system_of_ODEs

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

✓ Maple. Time used: 0.130 (sec). Leaf size: 44

ode:=[diff(x(t),t) = y(t)-z(t), diff(y(t),t) = x(t)^2+y(t), diff(z(t),t) = x(t)^2+z(t)]; 
dsolve(ode);
 

\begin{align*} \{x \left (t \right ) &= c_2 +c_3 \,{\mathrm e}^{t}\} \\ \{y \left (t \right ) &= \left (\int x \left (t \right )^{2} {\mathrm e}^{-t}d t +c_1 \right ) {\mathrm e}^{t}\} \\ \{z \left (t \right ) &= -\frac {d}{d t}x \left (t \right )+y \left (t \right )\} \\ \end{align*}

✓ Mathematica. Time used: 0.018 (sec). Leaf size: 109

ode={D[x[t],t]==y[t]-z[t],D[y[t],t]==x[t]^2+y[t],D[z[t],t]==x[t]^2+z[t]}; 
ic={}; 
DSolve[{ode,ic},{x[t],y[t],z[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)&\to e^{t-c_3}+c_1\\ y(t)&\to e^{t-c_3} \left (\int _1^{c_1+e^{t-c_3}}\frac {K[1]^2}{(c_1-K[1]){}^2}dK[1]+c_2\right )\\ z(t)&\to e^{t-c_3} \left (\int _1^{c_1+e^{t-c_3}}\frac {K[1]^2}{(c_1-K[1]){}^2}dK[1]-1+c_2\right ) \end{align*}

✗ Sympy

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
z = Function("z") 
ode=[Eq(-y(t) + z(t) + Derivative(x(t), t),0),Eq(-x(t)**2 - y(t) + Derivative(y(t), t),0),Eq(-x(t)**2 - z(t) + Derivative(z(t), t),0)] 
ics = {} 
dsolve(ode,func=[x(t),y(t),z(t)],ics=ics)
 

NotImplementedError :