12.21.13 problem section 10.4, problem 13

Internal problem ID [2251]
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 13
Date solved : Saturday, March 29, 2025 at 11:51:53 PM
CAS classification : system_of_ODEs

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

✓ Maple. Time used: 0.125 (sec). Leaf size: 65

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

\begin{align*} y_{1} \left (t \right ) &= c_1 \,{\mathrm e}^{4 t}+c_2 \,{\mathrm e}^{-4 t}+c_3 \,{\mathrm e}^{6 t} \\ y_{2} \left (t \right ) &= -\frac {c_2 \,{\mathrm e}^{-4 t}}{4}+c_3 \,{\mathrm e}^{6 t} \\ y_{3} \left (t \right ) &= -c_1 \,{\mathrm e}^{4 t}+\frac {c_2 \,{\mathrm e}^{-4 t}}{4}-c_3 \,{\mathrm e}^{6 t} \\ \end{align*}

✓ Mathematica. Time used: 0.008 (sec). Leaf size: 257

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

\begin{align*} \text {y1}(t)\to -\frac {e^{-\frac {1}{2} \left (\sqrt {73}-5\right ) t} \left (2 c_1 \left (\left (841 \sqrt {73}-7227\right ) e^{\sqrt {73} t}-7227-841 \sqrt {73}\right )+c_2 \left (\left (171 \sqrt {73}-1825\right ) e^{\sqrt {73} t}-1825-171 \sqrt {73}\right )\right )}{598016} \\ \text {y2}(t)\to \frac {e^{-\frac {1}{2} \left (\sqrt {73}-5\right ) t} \left (c_1 \left (\left (342 \sqrt {73}-3650\right ) e^{\sqrt {73} t}-3650-342 \sqrt {73}\right )-c_2 \left (\left (1971+143 \sqrt {73}\right ) e^{\sqrt {73} t}+1971-143 \sqrt {73}\right )\right )}{598016} \\ \text {y3}(t)\to \frac {e^{-\frac {1}{2} \left (\sqrt {73}-5\right ) t} \left (c_1 \left (\left (342 \sqrt {73}-3650\right ) e^{\sqrt {73} t}-3650-342 \sqrt {73}\right )-c_2 \left (\left (1971+143 \sqrt {73}\right ) e^{\sqrt {73} t}+1971-143 \sqrt {73}\right )\right )}{1196032} \\ \end{align*}

✓ Sympy. Time used: 0.155 (sec). Leaf size: 63

from sympy import * 
t = symbols("t") 
y__1 = Function("y__1") 
y__2 = Function("y__2") 
y__3 = Function("y__3") 
ode=[Eq(2*y__1(t) - 2*y__2(t) + 6*y__3(t) + Derivative(y__1(t), t),0),Eq(-2*y__1(t) - 6*y__2(t) - 2*y__3(t) + Derivative(y__2(t), t),0),Eq(2*y__1(t) + 2*y__2(t) - 2*y__3(t) + Derivative(y__3(t), t),0)] 
ics = {} 
dsolve(ode,func=[y__1(t),y__2(t),y__3(t)],ics=ics)
 

\[ \left [ y^{1}{\left (t \right )} = 4 C_{1} e^{- 4 t} - C_{2} e^{4 t} - C_{3} e^{6 t}, \ y^{2}{\left (t \right )} = - C_{1} e^{- 4 t} - C_{3} e^{6 t}, \ y^{3}{\left (t \right )} = C_{1} e^{- 4 t} + C_{2} e^{4 t} + C_{3} e^{6 t}\right ] \]