52.10.12 problem 11

Internal problem ID [8406]
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 : 11
Date solved : Sunday, March 30, 2025 at 01:00:58 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )&=-x \left (t \right )-y \left (t \right )\\ \frac {d}{d t}y \left (t \right )&=\frac {3 x \left (t \right )}{4}-\frac {3 y \left (t \right )}{2}+3 z \left (t \right )\\ \frac {d}{d t}z \left (t \right )&=\frac {x \left (t \right )}{8}+\frac {y \left (t \right )}{4}-\frac {z \left (t \right )}{2} \end{align*}

✓ Maple. Time used: 0.124 (sec). Leaf size: 66

ode:=[diff(x(t),t) = -x(t)-y(t), diff(y(t),t) = 3/4*x(t)-3/2*y(t)+3*z(t), diff(z(t),t) = 1/8*x(t)+1/4*y(t)-1/2*z(t)]; 
dsolve(ode);
 

\begin{align*} x \left (t \right ) &= c_1 \,{\mathrm e}^{-t}+c_2 \,{\mathrm e}^{-\frac {3 t}{2}}+c_3 \,{\mathrm e}^{-\frac {t}{2}} \\ y \left (t \right ) &= \frac {c_2 \,{\mathrm e}^{-\frac {3 t}{2}}}{2}-\frac {c_3 \,{\mathrm e}^{-\frac {t}{2}}}{2} \\ z \left (t \right ) &= -\frac {c_1 \,{\mathrm e}^{-t}}{4}-\frac {c_2 \,{\mathrm e}^{-\frac {3 t}{2}}}{4}-\frac {5 c_3 \,{\mathrm e}^{-\frac {t}{2}}}{12} \\ \end{align*}

✓ Mathematica. Time used: 0.006 (sec). Leaf size: 168

ode={D[x[t],t]==-x[t]-y[t],D[y[t],t]==3/4*x[t]-3/2*y[t]+3*z[t],D[z[t],t]==1/8x[t]+1/4*y[t]-1/2*z[t]}; 
ic={}; 
DSolve[{ode,ic},{x[t],y[t],z[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)\to \frac {1}{2} e^{-3 t/2} \left (c_1 \left (8 e^{t/2}-3 e^t-3\right )-4 \left (e^{t/2}-1\right ) \left (3 c_3 \left (e^{t/2}-1\right )+c_2\right )\right ) \\ y(t)\to \frac {1}{4} e^{-3 t/2} \left (3 c_1 \left (e^t-1\right )+4 \left (3 c_3 \left (e^t-1\right )+c_2\right )\right ) \\ z(t)\to \frac {1}{8} e^{-3 t/2} \left (c_1 \left (-8 e^{t/2}+5 e^t+3\right )+4 c_2 \left (e^{t/2}-1\right )+4 c_3 \left (-6 e^{t/2}+5 e^t+3\right )\right ) \\ \end{align*}

✓ Sympy. Time used: 0.156 (sec). Leaf size: 75

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
z = Function("z") 
ode=[Eq(x(t) + y(t) + Derivative(x(t), t),0),Eq(-3*x(t)/4 + 3*y(t)/2 - 3*z(t) + Derivative(y(t), t),0),Eq(-x(t)/8 - y(t)/4 + z(t)/2 + Derivative(z(t), t),0)] 
ics = {} 
dsolve(ode,func=[x(t),y(t),z(t)],ics=ics)
 

\[ \left [ x{\left (t \right )} = - 4 C_{1} e^{- \frac {3 t}{2}} - 4 C_{2} e^{- t} - \frac {12 C_{3} e^{- \frac {t}{2}}}{5}, \ y{\left (t \right )} = - 2 C_{1} e^{- \frac {3 t}{2}} + \frac {6 C_{3} e^{- \frac {t}{2}}}{5}, \ z{\left (t \right )} = C_{1} e^{- \frac {3 t}{2}} + C_{2} e^{- t} + C_{3} e^{- \frac {t}{2}}\right ] \]