52.10.26 problem 27

Internal problem ID [8420]
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 : 27
Date solved : Sunday, March 30, 2025 at 01:04:27 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )&=x \left (t \right )\\ \frac {d}{d t}y \left (t \right )&=2 x \left (t \right )+2 y \left (t \right )-z \left (t \right )\\ \frac {d}{d t}z \left (t \right )&=y \left (t \right ) \end{align*}

✓ Maple. Time used: 0.201 (sec). Leaf size: 45

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

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

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 65

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

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

✓ Sympy. Time used: 0.123 (sec). Leaf size: 65

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
z = Function("z") 
ode=[Eq(-x(t) + Derivative(x(t), t),0),Eq(-2*x(t) - 2*y(t) + z(t) + Derivative(y(t), t),0),Eq(-y(t) + Derivative(z(t), t),0)] 
ics = {} 
dsolve(ode,func=[x(t),y(t),z(t)],ics=ics)
 

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