52.10.10 problem 10

Internal problem ID [8404]
Book : DIFFERENTIAL EQUATIONS with Boundary Value Problems. DENNIS G. ZILL, WARREN S. WRIGHT, MICHAEL R. CULLEN. Brooks/Cole. Boston, MA. 2013. 8th edition.
Section : CHAPTER 8 SYSTEMS OF LINEAR FIRST-ORDER DIFFERENTIAL EQUATIONS. EXERCISES 8.2. Page 346
Problem number : 10
Date solved : Sunday, March 30, 2025 at 01:00:55 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )&=x \left (t \right )+z \left (t \right )\\ \frac {d}{d t}y \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: 0.135 (sec). Leaf size: 32

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

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

✓ Mathematica. Time used: 0.023 (sec). Leaf size: 128

ode={D[x[t],t]==x[t]+z[t],D[y[t],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 \frac {1}{2} \left (c_1 \left (e^{2 t}+1\right )+c_2 \left (e^{2 t}-1\right )\right ) \\ z(t)\to \frac {1}{2} \left (c_1 \left (e^{2 t}-1\right )+c_2 \left (e^{2 t}+1\right )\right ) \\ y(t)\to c_3 e^t \\ x(t)\to \frac {1}{2} \left (c_1 \left (e^{2 t}+1\right )+c_2 \left (e^{2 t}-1\right )\right ) \\ z(t)\to \frac {1}{2} \left (c_1 \left (e^{2 t}-1\right )+c_2 \left (e^{2 t}+1\right )\right ) \\ y(t)\to 0 \\ \end{align*}

✓ Sympy. Time used: 0.090 (sec). Leaf size: 27

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
z = Function("z") 
ode=[Eq(-x(t) - z(t) + Derivative(x(t), t),0),Eq(-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)
 

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