72.14.6 problem 10

Internal problem ID [14790]
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 : 10
Date solved : Thursday, March 13, 2025 at 04:19:02 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 )\\ \frac {d}{d t}z \left (t \right )&=-z \left (t \right ) \end{align*}

✓ Maple. Time used: 0.079 (sec). Leaf size: 32

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

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

✓ Mathematica. Time used: 0.019 (sec). Leaf size: 72

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

\begin{align*} x(t)\to e^{-2 t} (c_2 t+c_1) \\ y(t)\to c_2 e^{-2 t} \\ z(t)\to c_3 e^{-t} \\ x(t)\to e^{-2 t} (c_2 t+c_1) \\ y(t)\to c_2 e^{-2 t} \\ z(t)\to 0 \\ \end{align*}

✓ Sympy. Time used: 0.093 (sec). Leaf size: 32

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

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