6.21.12 problem section 10.4, problem 12

Internal problem ID [2250]
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 12
Date solved : Tuesday, September 30, 2025 at 05:25:27 AM
CAS classification : system_of_ODEs

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

✓ Maple. Time used: 0.123 (sec). Leaf size: 70

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

\begin{align*} y_{1} \left (t \right ) &= c_1 \,{\mathrm e}^{-t}+c_2 \,{\mathrm e}^{3 t}+c_3 \,{\mathrm e}^{-2 t} \\ y_{2} \left (t \right ) &= c_1 \,{\mathrm e}^{-t}+\frac {7 c_2 \,{\mathrm e}^{3 t}}{11}+2 c_3 \,{\mathrm e}^{-2 t} \\ y_{3} \left (t \right ) &= c_1 \,{\mathrm e}^{-t}+\frac {c_2 \,{\mathrm e}^{3 t}}{11}+c_3 \,{\mathrm e}^{-2 t} \\ \end{align*}

✓ Mathematica. Time used: 0.002 (sec). Leaf size: 102

ode={D[ y1[t],t]==4*y1[t]-1*y2[t]-4*y3[t],D[ y2[t],t]==4*y1[t]-3*y2[t]-2*y3[t],D[ y1[t],t]==1*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}{216} e^{-2 t} \left (c_1 \left (54 e^t-8\right )+c_2 \left (8-27 e^t\right )\right )\\ \text {y2}(t)&\to \frac {1}{216} e^{-2 t} \left (2 c_1 \left (27 e^t-8\right )+c_2 \left (16-27 e^t\right )\right )\\ \text {y3}(t)&\to \frac {1}{216} e^{-2 t} \left (c_1 \left (54 e^t-8\right )+c_2 \left (8-27 e^t\right )\right ) \end{align*}

✓ Sympy. Time used: 0.093 (sec). Leaf size: 66

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

\[ \left [ y^{1}{\left (t \right )} = C_{1} e^{- 2 t} + C_{2} e^{- t} + 11 C_{3} e^{3 t}, \ y^{2}{\left (t \right )} = 2 C_{1} e^{- 2 t} + C_{2} e^{- t} + 7 C_{3} e^{3 t}, \ y^{3}{\left (t \right )} = C_{1} e^{- 2 t} + C_{2} e^{- t} + C_{3} e^{3 t}\right ] \]