12.22.26 problem section 10.5, problem 26

Internal problem ID [2279]
Book : Elementary differential equations with boundary value problems. William F. Trench. Brooks/Cole 2001
Section : Chapter 10 Linear system of Differential equations. Section 10.5, constant coefficient homogeneous system II. Page 555
Problem number : section 10.5, problem 26
Date solved : Tuesday, March 04, 2025 at 01:52:56 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}y_{1} \left (t \right )&=-6 y_{1} \left (t \right )-4 y_{2} \left (t \right )-4 y_{3} \left (t \right )\\ \frac {d}{d t}y_{2} \left (t \right )&=2 y_{1} \left (t \right )-y_{2} \left (t \right )+y_{3} \left (t \right )\\ \frac {d}{d t}y_{3} \left (t \right )&=2 y_{1} \left (t \right )+3 y_{2} \left (t \right )+y_{3} \left (t \right ) \end{align*}

✓ Maple. Time used: 0.050 (sec). Leaf size: 72

ode:=[diff(y__1(t),t) = -6*y__1(t)-4*y__2(t)-4*y__3(t), diff(y__2(t),t) = 2*y__1(t)-y__2(t)+y__3(t), diff(y__3(t),t) = 2*y__1(t)+3*y__2(t)+y__3(t)]; 
dsolve(ode);
 

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

✓ Mathematica. Time used: 0.004 (sec). Leaf size: 100

ode={D[ y1[t],t]==-6*y1[t]-4*y2[t]-4*y3[t],D[ y2[t],t]==2*y1[t]-1*y2[t]+1*y3[t],D[ y3[t],t]==2*y1[t]+3*y2[t]+1*y3[t]}; 
ic={}; 
DSolve[{ode,ic},{y1[t],y2[t],y3[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} \text {y1}(t)\to e^{-2 t} (c_1 (1-4 t)-4 (c_2+c_3) t) \\ \text {y2}(t)\to e^{-2 t} \left (-2 (c_1+c_2+c_3) t^2+(2 c_1+c_2+c_3) t+c_2\right ) \\ \text {y3}(t)\to e^{-2 t} \left (2 (c_1+c_2+c_3) t^2+2 c_1 t+3 (c_2+c_3) t+c_3\right ) \\ \end{align*}

✓ Sympy. Time used: 0.158 (sec). Leaf size: 102

from sympy import * 
t = symbols("t") 
y__1 = Function("y__1") 
y__2 = Function("y__2") 
y__3 = Function("y__3") 
ode=[Eq(6*y__1(t) + 4*y__2(t) + 4*y__3(t) + Derivative(y__1(t), t),0),Eq(-2*y__1(t) + y__2(t) - y__3(t) + Derivative(y__2(t), t),0),Eq(-2*y__1(t) - 3*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)
 

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