6.22.21 problem section 10.5, problem 21

Internal problem ID [2274]
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 21
Date solved : Tuesday, September 30, 2025 at 05:25:42 AM
CAS classification : system_of_ODEs

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

With initial conditions

\begin{align*} y_{1} \left (0\right )&=-2 \\ y_{2} \left (0\right )&=1 \\ y_{3} \left (0\right )&=3 \\ \end{align*}

✓ Maple. Time used: 0.142 (sec). Leaf size: 62

ode:=[diff(y__1(t),t) = -y__1(t)-4*y__2(t)-y__3(t), diff(y__2(t),t) = 3*y__1(t)+6*y__2(t)+y__3(t), diff(y__3(t),t) = -3*y__1(t)-2*y__2(t)+3*y__3(t)]; 
ic:=[y__1(0) = -2, y__2(0) = 1, y__3(0) = 3]; 
dsolve([ode,op(ic)]);
 

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

✓ Mathematica. Time used: 0.004 (sec). Leaf size: 63

ode={D[ y1[t],t]==-1*y1[t]-4*y2[t]-1*y3[t],D[ y2[t],t]==3*y1[t]+6*y2[t]+1*y3[t],D[ y3[t],t]==-3*y1[t]-2*y2[t]+3*y3[t]}; 
ic={y1[0]==-2,y2[0]==1,y3[0]==3}; 
DSolve[{ode,ic},{y1[t],y2[t],y3[t]},t,IncludeSingularSolutions->True]
 

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

✓ Sympy. Time used: 0.110 (sec). Leaf size: 85

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) + 4*y__2(t) + y__3(t) + Derivative(y__1(t), t),0),Eq(-3*y__1(t) - 6*y__2(t) - y__3(t) + Derivative(y__2(t), t),0),Eq(3*y__1(t) + 2*y__2(t) - 3*y__3(t) + Derivative(y__3(t), t),0)] 
ics = {} 
dsolve(ode,func=[y__1(t),y__2(t),y__3(t)],ics=ics)
 

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