72.14.3 problem 5

Internal problem ID [14787]
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 : 5
Date solved : Thursday, March 13, 2025 at 04:18:59 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )&=-2 x \left (t \right )+3 y \left (t \right )\\ \frac {d}{d t}y \left (t \right )&=3 x \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: 0.059 (sec). Leaf size: 39

ode:=[diff(x(t),t) = -2*x(t)+3*y(t), diff(y(t),t) = 3*x(t)-2*y(t), diff(z(t),t) = -z(t)]; 
dsolve(ode);
 

\begin{align*} x \left (t \right ) &= {\mathrm e}^{-5 t} c_{1} +c_{2} {\mathrm e}^{t} \\ y &= -{\mathrm e}^{-5 t} c_{1} +c_{2} {\mathrm e}^{t} \\ z &= c_{3} {\mathrm e}^{-t} \\ \end{align*}

✓ Mathematica. Time used: 0.021 (sec). Leaf size: 150

ode={D[x[t],t]==-2*x[t]+3*y[t]+0*z[t],D[y[t],t]==3*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 \frac {1}{2} e^{-5 t} \left (c_1 \left (e^{6 t}+1\right )+c_2 \left (e^{6 t}-1\right )\right ) \\ y(t)\to \frac {1}{2} e^{-5 t} \left (c_1 \left (e^{6 t}-1\right )+c_2 \left (e^{6 t}+1\right )\right ) \\ z(t)\to c_3 e^{-t} \\ x(t)\to \frac {1}{2} e^{-5 t} \left (c_1 \left (e^{6 t}+1\right )+c_2 \left (e^{6 t}-1\right )\right ) \\ y(t)\to \frac {1}{2} e^{-5 t} \left (c_1 \left (e^{6 t}-1\right )+c_2 \left (e^{6 t}+1\right )\right ) \\ z(t)\to 0 \\ \end{align*}

✓ Sympy. Time used: 0.110 (sec). Leaf size: 34

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
z = Function("z") 
ode=[Eq(2*x(t) - 3*y(t) + Derivative(x(t), t),0),Eq(-3*x(t) + 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)
 

\[ \left [ x{\left (t \right )} = - C_{1} e^{- 5 t} + C_{2} e^{t}, \ y{\left (t \right )} = C_{1} e^{- 5 t} + C_{2} e^{t}, \ z{\left (t \right )} = C_{3} e^{- t}\right ] \]