78.5.21 problem 7.i

Internal problem ID [21060]
Book : A FIRST COURSE IN DIFFERENTIAL EQUATIONS FOR SCIENTISTS AND ENGINEERS. By Russell Herman. University of North Carolina Wilmington. LibreText. compiled on 06/09/2025
Section : Chapter 6, Linear systems. Problems section 6.9
Problem number : 7.i
Date solved : Thursday, October 02, 2025 at 07:01:52 PM
CAS classification : system_of_ODEs

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

✓ Maple. Time used: 0.178 (sec). Leaf size: 54

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

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

✓ Mathematica. Time used: 0.004 (sec). Leaf size: 129

ode={D[x[t],t]==5*x[t]+4*y[t]+2*z[t],D[y[t],t]==4*x[t]+5*y[t]+2*z[t],D[z[t],t]==2*x[t]+2*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}{9} e^t \left (c_1 \left (4 e^{9 t}+5\right )+2 (2 c_2+c_3) \left (e^{9 t}-1\right )\right )\\ y(t)&\to \frac {1}{9} e^t \left (4 c_1 \left (e^{9 t}-1\right )+c_2 \left (4 e^{9 t}+5\right )+2 c_3 \left (e^{9 t}-1\right )\right )\\ z(t)&\to \frac {1}{9} e^t \left (2 c_1 \left (e^{9 t}-1\right )+2 c_2 \left (e^{9 t}-1\right )+c_3 \left (e^{9 t}+8\right )\right ) \end{align*}

✓ Sympy. Time used: 0.075 (sec). Leaf size: 48

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

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