66.14.1 problem 1

Internal problem ID [16151]
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 : 1
Date solved : Thursday, October 02, 2025 at 10:42:59 AM
CAS classification : system_of_ODEs

\begin{align*} x^{\prime }\left (t \right )&=\frac {y}{10}\\ y^{\prime }&=\frac {z \left (t \right )}{5}\\ z^{\prime }\left (t \right )&=\frac {2 x \left (t \right )}{5} \end{align*}

✓ Maple. Time used: 0.155 (sec). Leaf size: 182

ode:=[diff(x(t),t) = 1/10*y(t), diff(y(t),t) = 1/5*z(t), diff(z(t),t) = 2/5*x(t)]; 
dsolve(ode);
 

\begin{align*} x \left (t \right ) &= \frac {c_1 \,{\mathrm e}^{\frac {t}{5}}}{2}-\frac {c_2 \,{\mathrm e}^{-\frac {t}{10}} \sin \left (\frac {\sqrt {3}\, t}{10}\right )}{4}+\frac {c_2 \,{\mathrm e}^{-\frac {t}{10}} \sqrt {3}\, \cos \left (\frac {\sqrt {3}\, t}{10}\right )}{4}-\frac {c_3 \,{\mathrm e}^{-\frac {t}{10}} \cos \left (\frac {\sqrt {3}\, t}{10}\right )}{4}-\frac {c_3 \,{\mathrm e}^{-\frac {t}{10}} \sqrt {3}\, \sin \left (\frac {\sqrt {3}\, t}{10}\right )}{4} \\ y \left (t \right ) &= c_1 \,{\mathrm e}^{\frac {t}{5}}-\frac {c_2 \,{\mathrm e}^{-\frac {t}{10}} \sin \left (\frac {\sqrt {3}\, t}{10}\right )}{2}-\frac {c_2 \,{\mathrm e}^{-\frac {t}{10}} \sqrt {3}\, \cos \left (\frac {\sqrt {3}\, t}{10}\right )}{2}-\frac {c_3 \,{\mathrm e}^{-\frac {t}{10}} \cos \left (\frac {\sqrt {3}\, t}{10}\right )}{2}+\frac {c_3 \,{\mathrm e}^{-\frac {t}{10}} \sqrt {3}\, \sin \left (\frac {\sqrt {3}\, t}{10}\right )}{2} \\ z \left (t \right ) &= c_1 \,{\mathrm e}^{\frac {t}{5}}+c_2 \,{\mathrm e}^{-\frac {t}{10}} \sin \left (\frac {\sqrt {3}\, t}{10}\right )+c_3 \,{\mathrm e}^{-\frac {t}{10}} \cos \left (\frac {\sqrt {3}\, t}{10}\right ) \\ \end{align*}

✓ Mathematica. Time used: 0.019 (sec). Leaf size: 269

ode={D[x[t],t]==0*x[t]+1/10*y[t]+0*z[t],D[y[t],t]==0*x[t]+0*y[t]+2/10*z[t],D[z[t],t]==4/10*x[t]+0*y[t]+0*z[t]}; 
ic={}; 
DSolve[{ode,ic},{x[t],y[t],z[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)&\to \frac {1}{6} e^{-t/10} \left ((2 c_1+c_2+c_3) e^{t/10} \sqrt [5]{e^t}+(4 c_1-c_2-c_3) \cos \left (\frac {\sqrt {3} t}{10}\right )+\sqrt {3} (c_2-c_3) \sin \left (\frac {\sqrt {3} t}{10}\right )\right )\\ y(t)&\to \frac {1}{3} e^{-t/10} \left ((2 c_1+c_2+c_3) e^{t/10} \sqrt [5]{e^t}-(2 c_1-2 c_2+c_3) \cos \left (\frac {\sqrt {3} t}{10}\right )-\sqrt {3} (2 c_1-c_3) \sin \left (\frac {\sqrt {3} t}{10}\right )\right )\\ z(t)&\to \frac {1}{3} e^{-t/10} \left ((2 c_1+c_2+c_3) e^{t/10} \sqrt [5]{e^t}-(2 c_1+c_2-2 c_3) \cos \left (\frac {\sqrt {3} t}{10}\right )+\sqrt {3} (2 c_1-c_2) \sin \left (\frac {\sqrt {3} t}{10}\right )\right ) \end{align*}

✓ Sympy. Time used: 0.187 (sec). Leaf size: 170

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
z = Function("z") 
ode=[Eq(-y(t)/10 + Derivative(x(t), t),0),Eq(-z(t)/5 + Derivative(y(t), t),0),Eq(-2*x(t)/5 + Derivative(z(t), t),0)] 
ics = {} 
dsolve(ode,func=[x(t),y(t),z(t)],ics=ics)
 

\[ \left [ x{\left (t \right )} = \frac {C_{3} e^{\frac {t}{5}}}{2} - \left (\frac {C_{1}}{4} + \frac {\sqrt {3} C_{2}}{4}\right ) e^{- \frac {t}{10}} \cos {\left (\frac {\sqrt {3} t}{10} \right )} - \left (\frac {\sqrt {3} C_{1}}{4} - \frac {C_{2}}{4}\right ) e^{- \frac {t}{10}} \sin {\left (\frac {\sqrt {3} t}{10} \right )}, \ y{\left (t \right )} = C_{3} e^{\frac {t}{5}} - \left (\frac {C_{1}}{2} - \frac {\sqrt {3} C_{2}}{2}\right ) e^{- \frac {t}{10}} \cos {\left (\frac {\sqrt {3} t}{10} \right )} + \left (\frac {\sqrt {3} C_{1}}{2} + \frac {C_{2}}{2}\right ) e^{- \frac {t}{10}} \sin {\left (\frac {\sqrt {3} t}{10} \right )}, \ z{\left (t \right )} = C_{1} e^{- \frac {t}{10}} \cos {\left (\frac {\sqrt {3} t}{10} \right )} - C_{2} e^{- \frac {t}{10}} \sin {\left (\frac {\sqrt {3} t}{10} \right )} + C_{3} e^{\frac {t}{5}}\right ] \]