66.14.4 problem 6

Internal problem ID [16154]
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 : 6
Date solved : Thursday, October 02, 2025 at 10:43:01 AM
CAS classification : system_of_ODEs

\begin{align*} x^{\prime }\left (t \right )&=x \left (t \right )+3 z \left (t \right )\\ y^{\prime }&=-y\\ z^{\prime }\left (t \right )&=-3 x \left (t \right )+z \left (t \right ) \end{align*}

✓ Maple. Time used: 0.147 (sec). Leaf size: 50

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

\begin{align*} x \left (t \right ) &= {\mathrm e}^{t} \left (\sin \left (3 t \right ) c_1 +\cos \left (3 t \right ) c_2 \right ) \\ y \left (t \right ) &= c_3 \,{\mathrm e}^{-t} \\ z \left (t \right ) &= -{\mathrm e}^{t} \left (\sin \left (3 t \right ) c_2 -\cos \left (3 t \right ) c_1 \right ) \\ \end{align*}

✓ Mathematica. Time used: 0.018 (sec). Leaf size: 108

ode={D[x[t],t]==1*x[t]+0*y[t]+3*z[t],D[y[t],t]==0*x[t]-1*y[t]+0*z[t],D[z[t],t]==-3*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^t (c_1 \cos (3 t)+c_2 \sin (3 t))\\ z(t)&\to e^t (c_2 \cos (3 t)-c_1 \sin (3 t))\\ y(t)&\to c_3 e^{-t}\\ x(t)&\to e^t (c_1 \cos (3 t)+c_2 \sin (3 t))\\ z(t)&\to e^t (c_2 \cos (3 t)-c_1 \sin (3 t))\\ y(t)&\to 0 \end{align*}

✓ Sympy. Time used: 0.056 (sec). Leaf size: 51

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
z = Function("z") 
ode=[Eq(-x(t) - 3*z(t) + Derivative(x(t), t),0),Eq(y(t) + Derivative(y(t), t),0),Eq(3*x(t) - z(t) + Derivative(z(t), t),0)] 
ics = {} 
dsolve(ode,func=[x(t),y(t),z(t)],ics=ics)
 

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