12.21.11 problem section 10.4, problem 11

Internal problem ID [2249]
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 11
Date solved : Tuesday, March 04, 2025 at 01:52:25 PM
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 )&=12 y_{1} \left (t \right )-4 y_{2} \left (t \right )+10 y_{3} \left (t \right )\\ \frac {d}{d t}y_{3} \left (t \right )&=-6 y_{1} \left (t \right )+y_{2} \left (t \right )-7 y_{3} \left (t \right ) \end{align*}

✓ Maple. Time used: 0.036 (sec). Leaf size: 72

ode:=[diff(y__1(t),t) = y__1(t)-y__2(t)+2*y__3(t), diff(y__2(t),t) = 12*y__1(t)-4*y__2(t)+10*y__3(t), diff(y__3(t),t) = -6*y__1(t)+y__2(t)-7*y__3(t)]; 
dsolve(ode);
 

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

✓ Mathematica. Time used: 0.018 (sec). Leaf size: 193

ode={D[ y1[t],t]==1*y1[t]-1*y2[t]+2*y3[t],D[ y2[t],t]==12*y1[t]-4*y2[t]+10*y3[t],D[ y1[t],t]==-6*y1[t]+1*y2[t]-7*y3[t]}; 
ic={}; 
DSolve[{ode,ic},{y1[t],y2[t],y3[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} \text {y1}(t)\to \frac {e^{-7 t/6} \left (71 (77 c_1-109 c_2) \cos \left (\frac {\sqrt {71} t}{6}\right )+\sqrt {71} (143 c_2-2479 c_1) \sin \left (\frac {\sqrt {71} t}{6}\right )\right )}{340800} \\ \text {y2}(t)\to \frac {e^{-7 t/6} \left (71 (2071 c_1-407 c_2) \cos \left (\frac {\sqrt {71} t}{6}\right )-\sqrt {71} (2717 c_1+5411 c_2) \sin \left (\frac {\sqrt {71} t}{6}\right )\right )}{852000} \\ \text {y3}(t)\to \frac {e^{-7 t/6} \left (639 (23 c_1+9 c_2) \cos \left (\frac {\sqrt {71} t}{6}\right )+3 \sqrt {71} (937 c_1-329 c_2) \sin \left (\frac {\sqrt {71} t}{6}\right )\right )}{568000} \\ \end{align*}

✓ Sympy. Time used: 0.158 (sec). Leaf size: 76

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(-12*y__1(t) + 4*y__2(t) - 10*y__3(t) + Derivative(y__2(t), t),0),Eq(6*y__1(t) - y__2(t) + 7*y__3(t) + Derivative(y__3(t), t),0)] 
ics = {} 
dsolve(ode,func=[y__1(t),y__2(t),y__3(t)],ics=ics)
 

\[ \left [ y^{1}{\left (t \right )} = - \frac {2 C_{1} e^{- 5 t}}{3} - C_{2} e^{- 3 t} - C_{3} e^{- 2 t}, \ y^{2}{\left (t \right )} = - 2 C_{1} e^{- 5 t} - 2 C_{2} e^{- 3 t} - C_{3} e^{- 2 t}, \ y^{3}{\left (t \right )} = C_{1} e^{- 5 t} + C_{2} e^{- 3 t} + C_{3} e^{- 2 t}\right ] \]