52.10.24 problem 25

Internal problem ID [8418]
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 : 25
Date solved : Sunday, March 30, 2025 at 01:04:24 PM
CAS classification : system_of_ODEs

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

✓ Maple. Time used: 0.160 (sec). Leaf size: 62

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

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

✓ Mathematica. Time used: 0.01 (sec). Leaf size: 141

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

\begin{align*} x(t)\to \frac {1}{25} \left (c_1 \left (e^{5 t} (29-20 t)-4\right )-4 \left (5 c_2 \left (e^{5 t}-1\right )+2 c_3 \left (e^{5 t} (5 t-1)+1\right )\right )\right ) \\ y(t)\to \frac {1}{5} c_1 \left (e^{5 t}-1\right )+\frac {2}{5} c_3 \left (e^{5 t}-1\right )+c_2 \\ z(t)\to \frac {1}{25} \left (2 c_1 \left (e^{5 t} (5 t-1)+1\right )+10 c_2 \left (e^{5 t}-1\right )+c_3 \left (e^{5 t} (20 t+21)+4\right )\right ) \\ \end{align*}

✓ Sympy. Time used: 0.159 (sec). Leaf size: 65

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

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