72.14.10 problem 14

Internal problem ID [14794]
Book : DIFFERENTIAL EQUATIONS by Paul Blanchard, Robert L. Devaney, Glen R. Hall. 4th edition. Brooks/Cole. Boston, USA. 2012
Section : Chapter 3. Linear Systems. Exercises section 3.8 page 371
Problem number : 14
Date solved : Thursday, March 13, 2025 at 04:19:05 AM
CAS classification : system_of_ODEs

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

✓ Maple. Time used: 0.152 (sec). Leaf size: 45

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

\begin{align*} x \left (t \right ) &= \frac {\left (c_{3} t^{2}+2 c_{2} t +2 c_{1} \right ) {\mathrm e}^{-2 t}}{2} \\ y &= \left (c_{3} t +c_{2} \right ) {\mathrm e}^{-2 t} \\ z &= c_{3} {\mathrm e}^{-2 t} \\ \end{align*}

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 57

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

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

✓ Sympy. Time used: 0.116 (sec). Leaf size: 54

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

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