72.14.5 problem 7

Internal problem ID [14789]
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 : 7
Date solved : Thursday, March 13, 2025 at 04:19:01 AM
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 y \left (t \right )-z \left (t \right )\\ \frac {d}{d t}z \left (t \right )&=-y \left (t \right )+2 z \left (t \right ) \end{align*}

✓ Maple. Time used: 0.072 (sec). Leaf size: 37

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

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

✓ Mathematica. Time used: 0.02 (sec). Leaf size: 144

ode={D[x[t],t]==1*x[t]+0*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]-1*y[t]+2*z[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 \frac {1}{2} e^t \left (c_2 e^{2 t}-c_3 e^{2 t}+c_2+c_3\right ) \\ z(t)\to \frac {1}{2} e^t \left (c_2 \left (-e^{2 t}\right )+c_3 e^{2 t}+c_2+c_3\right ) \\ x(t)\to 0 \\ y(t)\to \frac {1}{2} e^t \left (c_2 e^{2 t}-c_3 e^{2 t}+c_2+c_3\right ) \\ z(t)\to \frac {1}{2} e^t \left (c_2 \left (-e^{2 t}\right )+c_3 e^{2 t}+c_2+c_3\right ) \\ \end{align*}

✓ Sympy. Time used: 0.100 (sec). Leaf size: 34

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

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