52.10.29 problem 30

Internal problem ID [8423]
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 : 30
Date solved : Sunday, March 30, 2025 at 01:04:31 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}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 ) \end{align*}

With initial conditions

\begin{align*} x \left (0\right ) = 1\\ y \left (0\right ) = 2\\ z \left (0\right ) = 5 \end{align*}

✓ Maple. Time used: 0.149 (sec). Leaf size: 36

ode:=[diff(x(t),t) = z(t), diff(y(t),t) = y(t), diff(z(t),t) = x(t)]; 
ic:=x(0) = 1y(0) = 2z(0) = 5; 
dsolve([ode,ic]);
 

\begin{align*} x \left (t \right ) &= 3 \,{\mathrm e}^{t}-2 \,{\mathrm e}^{-t} \\ y \left (t \right ) &= 2 \,{\mathrm e}^{t} \\ z \left (t \right ) &= 3 \,{\mathrm e}^{t}+2 \,{\mathrm e}^{-t} \\ \end{align*}

✓ Mathematica. Time used: 0.024 (sec). Leaf size: 42

ode={D[x[t],t]==z[t],D[y[t],t]==y[t],D[z[t],t]==x[t]}; 
ic={x[0]==1,y[0]==2,z[0]==5}; 
DSolve[{ode,ic},{x[t],y[t],z[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)\to 3 e^t-2 e^{-t} \\ z(t)\to 2 e^{-t}+3 e^t \\ y(t)\to 2 e^t \\ \end{align*}

✓ Sympy. Time used: 0.094 (sec). Leaf size: 31

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

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