52.10.8 problem 8

Internal problem ID [8402]
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 : 8
Date solved : Wednesday, March 05, 2025 at 05:44:41 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )&=2 x \left (t \right )-7 y \left (t \right )\\ \frac {d}{d t}y \left (t \right )&=5 x \left (t \right )+10 y \left (t \right )+4 z \left (t \right )\\ \frac {d}{d t}z \left (t \right )&=5 y \left (t \right )+2 z \left (t \right ) \end{align*}

✓ Maple. Time used: 0.044 (sec). Leaf size: 65

ode:=[diff(x(t),t) = 2*x(t)-7*y(t), diff(y(t),t) = 5*x(t)+10*y(t)+4*z(t), diff(z(t),t) = 5*y(t)+2*z(t)]; 
dsolve(ode);
 

\begin{align*} x \left (t \right ) &= -\frac {7 c_{2} {\mathrm e}^{7 t}}{5}-\frac {7 c_3 \,{\mathrm e}^{5 t}}{3}+c_{1} {\mathrm e}^{2 t} \\ y &= c_{2} {\mathrm e}^{7 t}+c_3 \,{\mathrm e}^{5 t} \\ z \left (t \right ) &= c_{2} {\mathrm e}^{7 t}+\frac {5 c_3 \,{\mathrm e}^{5 t}}{3}-\frac {5 c_{1} {\mathrm e}^{2 t}}{4} \\ \end{align*}

✓ Mathematica. Time used: 0.005 (sec). Leaf size: 190

ode={D[x[t],t]==2*x[t]-7*y[t],D[y[t],t]==5*x[t]+10*y[t]+4*z[t],D[z[t],t]==5*y[t]+2*z[t]}; 
ic={}; 
DSolve[{ode,ic},{x[t],y[t],z[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)\to -\frac {1}{30} e^{2 t} \left (5 c_1 \left (-35 e^{3 t}+21 e^{5 t}+8\right )+7 \left (-5 (3 c_2+4 c_3) e^{3 t}+3 (5 c_2+4 c_3) e^{5 t}+8 c_3\right )\right ) \\ y(t)\to \frac {1}{2} e^{5 t} \left (5 c_1 \left (e^{2 t}-1\right )+c_2 \left (5 e^{2 t}-3\right )+4 c_3 \left (e^{2 t}-1\right )\right ) \\ z(t)\to \frac {1}{6} e^{2 t} \left (5 c_1 \left (-5 e^{3 t}+3 e^{5 t}+2\right )-5 (3 c_2+4 c_3) e^{3 t}+3 (5 c_2+4 c_3) e^{5 t}+14 c_3\right ) \\ \end{align*}

✓ Sympy. Time used: 0.151 (sec). Leaf size: 75

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
z = Function("z") 
ode=[Eq(-2*x(t) + 7*y(t) + Derivative(x(t), t),0),Eq(-5*x(t) - 10*y(t) - 4*z(t) + Derivative(y(t), t),0),Eq(-5*y(t) - 2*z(t) + Derivative(z(t), t),0)] 
ics = {} 
dsolve(ode,func=[x(t),y(t),z(t)],ics=ics)
 

\[ \left [ x{\left (t \right )} = - \frac {4 C_{1} e^{2 t}}{5} - \frac {7 C_{2} e^{5 t}}{5} - \frac {7 C_{3} e^{7 t}}{5}, \ y{\left (t \right )} = \frac {3 C_{2} e^{5 t}}{5} + C_{3} e^{7 t}, \ z{\left (t \right )} = C_{1} e^{2 t} + C_{2} e^{5 t} + C_{3} e^{7 t}\right ] \]