6.21.8 problem section 10.4, problem 8

Internal problem ID [2246]
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 8
Date solved : Tuesday, September 30, 2025 at 05:25:24 AM
CAS classification : system_of_ODEs

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

✓ Maple. Time used: 0.125 (sec). Leaf size: 71

ode:=[diff(y__1(t),t) = y__1(t)-y__2(t)-2*y__3(t), diff(y__2(t),t) = y__1(t)-2*y__2(t)-3*y__3(t), diff(y__3(t),t) = -4*y__1(t)+y__2(t)-y__3(t)]; 
dsolve(ode);
 

\begin{align*} y_{1} \left (t \right ) &= c_1 \,{\mathrm e}^{-3 t}+c_2 \,{\mathrm e}^{-t}+c_3 \,{\mathrm e}^{2 t} \\ y_{2} \left (t \right ) &= 2 c_1 \,{\mathrm e}^{-3 t}+4 c_2 \,{\mathrm e}^{-t}+c_3 \,{\mathrm e}^{2 t} \\ y_{3} \left (t \right ) &= c_1 \,{\mathrm e}^{-3 t}-c_2 \,{\mathrm e}^{-t}-c_3 \,{\mathrm e}^{2 t} \\ \end{align*}

✓ Mathematica. Time used: 0.015 (sec). Leaf size: 102

ode={D[ y1[t],t]==1*y1[t]-1*y2[t]-2*y3[t],D[ y2[t],t]==1*y1[t]-2*y2[t]-3*y3[t],D[ y1[t],t]==-4*y1[t]+1*y2[t]-1*y3[t]}; 
ic={}; 
DSolve[{ode,ic},{y1[t],y2[t],y3[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} \text {y1}(t)&\to \frac {1}{576} e^{-3 t} \left (c_1 \left (63-128 e^t\right )+c_2 \left (64 e^t-27\right )\right )\\ \text {y2}(t)&\to \frac {1}{864} e^{-3 t} \left (c_2 \left (224 e^t-81\right )-7 c_1 \left (64 e^t-27\right )\right )\\ \text {y3}(t)&\to \frac {e^{-3 t} \left (c_1 \left (189-128 e^t\right )+c_2 \left (64 e^t-81\right )\right )}{1728} \end{align*}

✓ Sympy. Time used: 0.095 (sec). Leaf size: 65

from sympy import * 
t = symbols("t") 
y__1 = Function("y__1") 
y__2 = Function("y__2") 
y__3 = Function("y__3") 
ode=[Eq(-y__1(t) + y__2(t) + 2*y__3(t) + Derivative(y__1(t), t),0),Eq(-y__1(t) + 2*y__2(t) + 3*y__3(t) + Derivative(y__2(t), t),0),Eq(4*y__1(t) - y__2(t) + y__3(t) + Derivative(y__3(t), t),0)] 
ics = {} 
dsolve(ode,func=[y__1(t),y__2(t),y__3(t)],ics=ics)
 

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