12.22.30 problem section 10.5, problem 30

Internal problem ID [2283]
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 30
Date solved : Tuesday, March 04, 2025 at 01:53:00 PM
CAS classification : system_of_ODEs

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

✓ Maple. Time used: 0.038 (sec). Leaf size: 43

ode:=[diff(y__1(t),t) = -4*y__1(t)-y__3(t), diff(y__2(t),t) = -y__1(t)-3*y__2(t)-y__3(t), diff(y__3(t),t) = y__1(t)-2*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 ) &= {\mathrm e}^{-3 t} \left (c_3 t +c_1 +c_2 \right ) \\ y_{3} \left (t \right ) &= -{\mathrm e}^{-3 t} \left (c_3 t +c_2 +c_3 \right ) \\ \end{align*}

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 63

ode={D[ y1[t],t]==-4*y1[t]-0*y2[t]-1*y3[t],D[ y2[t],t]==-1*y1[t]-3*y2[t]-1*y3[t],D[ y3[t],t]==1*y1[t]-0*y2[t]-2*y3[t]}; 
ic={}; 
DSolve[{ode,ic},{y1[t],y2[t],y3[t]},t,IncludeSingularSolutions->True]
 

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

✓ Sympy. Time used: 0.134 (sec). Leaf size: 58

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

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