60.9.46 problem 1901

Internal problem ID [11825]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 8, system of first order odes
Problem number : 1901
Date solved : Sunday, March 30, 2025 at 09:16:05 PM
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 )+y \left (t \right )\\ \frac {d}{d t}z \left (t \right )&=x \left (t \right )+z \left (t \right ) \end{align*}

✓ Maple. Time used: 0.155 (sec). Leaf size: 47

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

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

✓ Mathematica. Time used: 0.005 (sec). Leaf size: 93

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

\begin{align*} x(t)\to (c_2-c_3) \left (e^t-1\right )+c_1 \\ y(t)\to c_1 \left (e^t-1\right )+c_2 \left (e^t t+1\right )-c_3 \left (e^t (t-1)+1\right ) \\ z(t)\to c_1 \left (e^t-1\right )+c_2 \left (e^t (t-1)+1\right )-c_3 \left (e^t (t-2)+1\right ) \\ \end{align*}

✓ Sympy. Time used: 0.109 (sec). Leaf size: 41

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) - y(t) + Derivative(y(t), t),0),Eq(-x(t) - z(t) + Derivative(z(t), t),0)] 
ics = {} 
dsolve(ode,func=[x(t),y(t),z(t)],ics=ics)
 

\[ \left [ x{\left (t \right )} = - C_{1} + C_{2} e^{t}, \ y{\left (t \right )} = C_{1} + C_{2} t e^{t} + \left (C_{2} + C_{3}\right ) e^{t}, \ z{\left (t \right )} = C_{1} + C_{2} t e^{t} + C_{3} e^{t}\right ] \]