52.9.16 problem 16

Internal problem ID [8394]
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 8.1. Page 332
Problem number : 16
Date solved : Wednesday, March 05, 2025 at 05:44:33 AM
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 )&=x \left (t \right )+y \left (t \right )\\ \frac {d}{d t}z \left (t \right )&=-2 x \left (t \right )-z \left (t \right ) \end{align*}

✓ Maple. Time used: 0.101 (sec). Leaf size: 64

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

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

✓ Mathematica. Time used: 0.009 (sec). Leaf size: 76

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

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

✓ Sympy. Time used: 0.117 (sec). Leaf size: 54

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(-x(t) - y(t) + Derivative(y(t), t),0),Eq(2*x(t) + z(t) + Derivative(z(t), t),0)] 
ics = {} 
dsolve(ode,func=[x(t),y(t),z(t)],ics=ics)
 

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