52.10.38 problem 45

Internal problem ID [8432]
Book : DIFFERENTIAL EQUATIONS with Boundary Value Problems. DENNIS G. ZILL, WARREN S. WRIGHT, MICHAEL R. CULLEN. Brooks/Cole. Boston, MA. 2013. 8th edition.
Section : CHAPTER 8 SYSTEMS OF LINEAR FIRST-ORDER DIFFERENTIAL EQUATIONS. EXERCISES 8.2. Page 346
Problem number : 45
Date solved : Sunday, March 30, 2025 at 01:04:44 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )&=x \left (t \right )-12 y \left (t \right )-14 z \left (t \right )\\ \frac {d}{d t}y \left (t \right )&=x \left (t \right )+2 y \left (t \right )-3 z \left (t \right )\\ \frac {d}{d t}z \left (t \right )&=x \left (t \right )+y \left (t \right )-2 z \left (t \right ) \end{align*}

With initial conditions

\begin{align*} x \left (0\right ) = 4\\ y \left (0\right ) = 6\\ z \left (0\right ) = -7 \end{align*}

✓ Maple. Time used: 0.158 (sec). Leaf size: 61

ode:=[diff(x(t),t) = x(t)-12*y(t)-14*z(t), diff(y(t),t) = x(t)+2*y(t)-3*z(t), diff(z(t),t) = x(t)+y(t)-2*z(t)]; 
ic:=x(0) = 4y(0) = 6z(0) = -7; 
dsolve([ode,ic]);
 

\begin{align*} x \left (t \right ) &= -25 \,{\mathrm e}^{t}+11 \sin \left (5 t \right )+29 \cos \left (5 t \right ) \\ y \left (t \right ) &= 7 \,{\mathrm e}^{t}+6 \sin \left (5 t \right )-\cos \left (5 t \right ) \\ z \left (t \right ) &= -6 \,{\mathrm e}^{t}-\cos \left (5 t \right )+6 \sin \left (5 t \right ) \\ \end{align*}

✓ Mathematica. Time used: 0.013 (sec). Leaf size: 65

ode={D[x[t],t]==x[t]-12*y[t]-14*z[t],D[y[t],t]==x[t]+2*y[t]-3*z[t],D[z[t],t]==x[t]+y[t]-2*z[t]}; 
ic={x[0]==4,y[0]==6,z[0]==-7}; 
DSolve[{ode,ic},{x[t],y[t],z[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)\to -25 e^t+11 \sin (5 t)+29 \cos (5 t) \\ y(t)\to 7 e^t+6 \sin (5 t)-\cos (5 t) \\ z(t)\to -6 e^t+6 \sin (5 t)-\cos (5 t) \\ \end{align*}

✓ Sympy. Time used: 0.143 (sec). Leaf size: 75

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
z = Function("z") 
ode=[Eq(-x(t) + 12*y(t) + 14*z(t) + Derivative(x(t), t),0),Eq(-x(t) - 2*y(t) + 3*z(t) + Derivative(y(t), t),0),Eq(-x(t) - y(t) + 2*z(t) + Derivative(z(t), t),0)] 
ics = {} 
dsolve(ode,func=[x(t),y(t),z(t)],ics=ics)
 

\[ \left [ x{\left (t \right )} = \frac {25 C_{1} e^{t}}{6} + \left (C_{2} - 5 C_{3}\right ) \cos {\left (5 t \right )} - \left (5 C_{2} + C_{3}\right ) \sin {\left (5 t \right )}, \ y{\left (t \right )} = - \frac {7 C_{1} e^{t}}{6} + C_{2} \cos {\left (5 t \right )} - C_{3} \sin {\left (5 t \right )}, \ z{\left (t \right )} = C_{1} e^{t} + C_{2} \cos {\left (5 t \right )} - C_{3} \sin {\left (5 t \right )}\right ] \]