60.9.47 problem 1902

Internal problem ID [11826]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 8, system of first order odes
Problem number : 1902
Date solved : Wednesday, March 05, 2025 at 03:07:29 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )-y \left (t \right )+z \left (t \right )&=0\\ \frac {d}{d t}y \left (t \right )-x \left (t \right )-y \left (t \right )&=t\\ \frac {d}{d t}z \left (t \right )-x \left (t \right )-z \left (t \right )&=t \end{align*}

✓ Maple. Time used: 0.089 (sec). Leaf size: 55

ode:=[diff(x(t),t)-y(t)+z(t) = 0, diff(y(t),t)-x(t)-y(t) = t, diff(z(t),t)-x(t)-z(t) = t]; 
dsolve(ode);
 

\begin{align*} x \left (t \right ) &= c_{2} +c_3 \,{\mathrm e}^{t} \\ y \left (t \right ) &= c_3 \,{\mathrm e}^{t} t +c_{1} {\mathrm e}^{t}-c_{2} -t -1 \\ z &= c_3 \,{\mathrm e}^{t} t +c_{1} {\mathrm e}^{t}-c_3 \,{\mathrm e}^{t}-c_{2} -t -1 \\ \end{align*}

✓ Mathematica. Time used: 0.011 (sec). Leaf size: 246

ode={D[x[t],t]-y[t]+z[t]==0,D[y[t],t]-x[t]-y[t]==t,D[z[t],t]-x[t]-z[t]==t}; 
ic={}; 
DSolve[{ode,ic},{x[t],y[t],z[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)\to \left (e^t-1\right ) \int _1^te^{-K[1]} K[1]dK[1]-\left (e^t-1\right ) \int _1^te^{-K[2]} K[2]dK[2]+c_2 \left (e^t-1\right )-c_3 \left (e^t-1\right )+c_1 \\ y(t)\to \left (e^t t+1\right ) \int _1^te^{-K[1]} K[1]dK[1]+\left (-e^t (t-1)-1\right ) \int _1^te^{-K[2]} K[2]dK[2]+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 \left (e^t (t-1)+1\right ) \int _1^te^{-K[1]} K[1]dK[1]-\left (e^t (t-2)+1\right ) \int _1^te^{-K[2]} K[2]dK[2]+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: 0.145 (sec). Leaf size: 48

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

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