6.21.9 problem section 10.4, problem 9

Internal problem ID [2247]
Book : Elementary differential equations with boundary value problems. William F. Trench. Brooks/Cole 2001
Section : Chapter 10 Linear system of Differential equations. Section 10.4, constant coefficient homogeneous system. Page 540
Problem number : section 10.4, problem 9
Date solved : Tuesday, September 30, 2025 at 05:25:25 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}y_{1} \left (t \right )&=-6 y_{1} \left (t \right )-4 y_{2} \left (t \right )-8 y_{3} \left (t \right )\\ \frac {d}{d t}y_{2} \left (t \right )&=-4 y_{1} \left (t \right )-4 y_{3} \left (t \right )\\ \frac {d}{d t}y_{3} \left (t \right )&=-8 y_{1} \left (t \right )-4 y_{2} \left (t \right )-6 y_{3} \left (t \right ) \end{align*}

✓ Maple. Time used: 0.164 (sec). Leaf size: 66

ode:=[diff(y__1(t),t) = -6*y__1(t)-4*y__2(t)-8*y__3(t), diff(y__2(t),t) = -4*y__1(t)-4*y__3(t), diff(y__3(t),t) = -8*y__1(t)-4*y__2(t)-6*y__3(t)]; 
dsolve(ode);
 

\begin{align*} y_{1} \left (t \right ) &= 2 c_2 \,{\mathrm e}^{-16 t}+2 c_3 \,{\mathrm e}^{2 t}+{\mathrm e}^{2 t} c_1 \\ y_{2} \left (t \right ) &= c_2 \,{\mathrm e}^{-16 t}+c_3 \,{\mathrm e}^{2 t} \\ y_{3} \left (t \right ) &= 2 c_2 \,{\mathrm e}^{-16 t}-\frac {5 c_3 \,{\mathrm e}^{2 t}}{2}-{\mathrm e}^{2 t} c_1 \\ \end{align*}

✓ Mathematica. Time used: 0.002 (sec). Leaf size: 116

ode={D[ y1[t],t]==-6*y1[t]-4*y2[t]-8*y3[t],D[ y2[t],t]==-4*y1[t]-0*y2[t]-4*y3[t],D[ y1[t],t]==-8*y1[t]-4*y2[t]-6*y3[t]}; 
ic={}; 
DSolve[{ode,ic},{y1[t],y2[t],y3[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} \text {y1}(t)&\to \frac {e^{-16 t} \left (c_1 \left (8-4913 e^{18 t}\right )+2 c_2 \left (4913 e^{18 t}+1\right )\right )}{44217}\\ \text {y2}(t)&\to \frac {e^{-16 t} \left (4 c_1 \left (4913 e^{18 t}+1\right )+c_2 \left (1-39304 e^{18 t}\right )\right )}{44217}\\ \text {y3}(t)&\to \frac {e^{-16 t} \left (c_1 \left (8-4913 e^{18 t}\right )+2 c_2 \left (4913 e^{18 t}+1\right )\right )}{44217} \end{align*}

✓ Sympy. Time used: 0.094 (sec). Leaf size: 51

from sympy import * 
t = symbols("t") 
y__1 = Function("y__1") 
y__2 = Function("y__2") 
y__3 = Function("y__3") 
ode=[Eq(6*y__1(t) + 4*y__2(t) + 8*y__3(t) + Derivative(y__1(t), t),0),Eq(4*y__1(t) + 4*y__3(t) + Derivative(y__2(t), t),0),Eq(8*y__1(t) + 4*y__2(t) + 6*y__3(t) + Derivative(y__3(t), t),0)] 
ics = {} 
dsolve(ode,func=[y__1(t),y__2(t),y__3(t)],ics=ics)
 

\[ \left [ y^{1}{\left (t \right )} = C_{1} e^{- 16 t} - \left (C_{2} + \frac {C_{3}}{2}\right ) e^{2 t}, \ y^{2}{\left (t \right )} = \frac {C_{1} e^{- 16 t}}{2} + C_{3} e^{2 t}, \ y^{3}{\left (t \right )} = C_{1} e^{- 16 t} + C_{2} e^{2 t}\right ] \]