12.23.6 problem section 10.6, problem 6

Internal problem ID [2291]
Book : Elementary differential equations with boundary value problems. William F. Trench. Brooks/Cole 2001
Section : Chapter 10 Linear system of Differential equations. Section 10.6, constant coefficient homogeneous system III. Page 566
Problem number : section 10.6, problem 6
Date solved : Tuesday, March 04, 2025 at 01:53:17 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}y_{1} \left (t \right )&=-3 y_{1} \left (t \right )+3 y_{2} \left (t \right )+y_{3} \left (t \right )\\ \frac {d}{d t}y_{2} \left (t \right )&=y_{1} \left (t \right )-5 y_{2} \left (t \right )-3 y_{3} \left (t \right )\\ \frac {d}{d t}y_{3} \left (t \right )&=-3 y_{1} \left (t \right )+7 y_{2} \left (t \right )+3 y_{3} \left (t \right ) \end{align*}

✓ Maple. Time used: 0.041 (sec). Leaf size: 114

ode:=[diff(y__1(t),t) = -3*y__1(t)+3*y__2(t)+y__3(t), diff(y__2(t),t) = y__1(t)-5*y__2(t)-3*y__3(t), diff(y__3(t),t) = -3*y__1(t)+7*y__2(t)+3*y__3(t)]; 
dsolve(ode);
 

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

✓ Mathematica. Time used: 0.01 (sec). Leaf size: 158

ode={D[ y1[t],t]==-3*y1[t]+3*y2[t]+1*y3[t],D[ y2[t],t]==1*y1[t]-5*y2[t]-3*y3[t],D[ y3[t],t]==-3*y1[t]+7*y2[t]+3*y3[t]}; 
ic={}; 
DSolve[{ode,ic},{y1[t],y2[t],y3[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} \text {y1}(t)\to e^{-2 t} \left ((c_1-c_2-c_3) e^t+(c_2+c_3) \cos (2 t)+(-c_1+2 c_2+c_3) \sin (2 t)\right ) \\ \text {y2}(t)\to e^{-2 t} \left ((c_1-c_2-c_3) e^t+(-c_1+2 c_2+c_3) \cos (2 t)-(c_2+c_3) \sin (2 t)\right ) \\ \text {y3}(t)\to e^{-2 t} \left ((-c_1+c_2+c_3) e^t+(c_1-c_2) \cos (2 t)+(-c_1+3 c_2+2 c_3) \sin (2 t)\right ) \\ \end{align*}

✓ Sympy. Time used: 0.211 (sec). Leaf size: 114

from sympy import * 
t = symbols("t") 
y__1 = Function("y__1") 
y__2 = Function("y__2") 
y__3 = Function("y__3") 
ode=[Eq(3*y__1(t) - 3*y__2(t) - y__3(t) + Derivative(y__1(t), t),0),Eq(-y__1(t) + 5*y__2(t) + 3*y__3(t) + Derivative(y__2(t), t),0),Eq(3*y__1(t) - 7*y__2(t) - 3*y__3(t) + Derivative(y__3(t), t),0)] 
ics = {} 
dsolve(ode,func=[y__1(t),y__2(t),y__3(t)],ics=ics)
 

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