72.14.15 problem 20

Internal problem ID [14799]
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 : 20
Date solved : Thursday, March 13, 2025 at 04:19:11 AM
CAS classification : system_of_ODEs

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

✓ Maple. Time used: 0.066 (sec). Leaf size: 41

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

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

✓ Mathematica. Time used: 0.004 (sec). Leaf size: 94

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

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

✓ Sympy. Time used: 0.104 (sec). Leaf size: 32

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
z = Function("z") 
ode=[Eq(y(t) - z(t) + Derivative(x(t), t),0),Eq(x(t) - z(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)
 

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