12.22.29 problem section 10.5, problem 29

Internal problem ID [2282]
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 29
Date solved : Tuesday, March 04, 2025 at 01:52:59 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}y_{1} \left (t \right )&=-y_{1} \left (t \right )-12 y_{2} \left (t \right )+8 y_{3} \left (t \right )\\ \frac {d}{d t}y_{2} \left (t \right )&=y_{1} \left (t \right )-9 y_{2} \left (t \right )+4 y_{3} \left (t \right )\\ \frac {d}{d t}y_{3} \left (t \right )&=y_{1} \left (t \right )-6 y_{2} \left (t \right )+y_{3} \left (t \right ) \end{align*}

✓ Maple. Time used: 0.051 (sec). Leaf size: 52

ode:=[diff(y__1(t),t) = -y__1(t)-12*y__2(t)+8*y__3(t), diff(y__2(t),t) = y__1(t)-9*y__2(t)+4*y__3(t), diff(y__3(t),t) = y__1(t)-6*y__2(t)+y__3(t)]; 
dsolve(ode);
 

\begin{align*} y_{1} \left (t \right ) &= {\mathrm e}^{-3 t} \left (c_3 t +c_2 \right ) \\ y_{2} \left (t \right ) &= \frac {{\mathrm e}^{-3 t} \left (c_3 t +2 c_1 +c_2 \right )}{2} \\ y_{3} \left (t \right ) &= \frac {{\mathrm e}^{-3 t} \left (4 c_3 t +12 c_1 +4 c_2 +c_3 \right )}{8} \\ \end{align*}

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 79

ode={D[ y1[t],t]==-1*y1[t]-12*y2[t]+8*y3[t],D[ y2[t],t]==1*y1[t]-9*y2[t]+4*y3[t],D[ y3[t],t]==1*y1[t]-6*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^{-3 t} (2 c_1 t-12 c_2 t+8 c_3 t+c_1) \\ \text {y2}(t)\to e^{-3 t} ((c_1-6 c_2+4 c_3) t+c_2) \\ \text {y3}(t)\to e^{-3 t} ((c_1-6 c_2+4 c_3) t+c_3) \\ \end{align*}

✓ Sympy. Time used: 0.144 (sec). Leaf size: 61

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) + 12*y__2(t) - 8*y__3(t) + Derivative(y__1(t), t),0),Eq(-y__1(t) + 9*y__2(t) - 4*y__3(t) + Derivative(y__2(t), t),0),Eq(-y__1(t) + 6*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)
 

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