12.22.18 problem section 10.5, problem 18

Internal problem ID [2271]
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 18
Date solved : Tuesday, March 04, 2025 at 01:52:48 PM
CAS classification : system_of_ODEs

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

With initial conditions

\begin{align*} y_{1} \left (0\right ) = 6\\ y_{2} \left (0\right ) = 5\\ y_{3} \left (0\right ) = -7 \end{align*}

✓ Maple. Time used: 0.040 (sec). Leaf size: 56

ode:=[diff(y__1(t),t) = -y__1(t)+y__2(t), diff(y__2(t),t) = y__1(t)-y__2(t)-2*y__3(t), diff(y__3(t),t) = -y__1(t)-y__2(t)-y__3(t)]; 
ic:=y__1(0) = 6y__2(0) = 5y__3(0) = -7; 
dsolve([ode,ic]);
 

\begin{align*} y_{1} \left (t \right ) &= 4 \,{\mathrm e}^{t}+2 \,{\mathrm e}^{-2 t}-{\mathrm e}^{-2 t} t \\ y_{2} \left (t \right ) &= 8 \,{\mathrm e}^{t}-3 \,{\mathrm e}^{-2 t}+{\mathrm e}^{-2 t} t \\ y_{3} \left (t \right ) &= -6 \,{\mathrm e}^{t}-{\mathrm e}^{-2 t} \\ \end{align*}

✓ Mathematica. Time used: 0.009 (sec). Leaf size: 58

ode={D[ y1[t],t]==-1*y1[t]+1*y2[t]+0*y3[t],D[ y2[t],t]==1*y1[t]-1*y2[t]-2*y3[t],D[ y3[t],t]==-1*y1[t]-1*y2[t]-1*y3[t]}; 
ic={y1[0]==6,y2[0]==5,y3[0]==-7}; 
DSolve[{ode,ic},{y1[t],y2[t],y3[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} \text {y1}(t)\to e^{-2 t} \left (-t+4 e^{3 t}+2\right ) \\ \text {y2}(t)\to e^{-2 t} \left (t+8 e^{3 t}-3\right ) \\ \text {y3}(t)\to -e^{-2 t}-6 e^t \\ \end{align*}

✓ Sympy. Time used: 0.155 (sec). Leaf size: 68

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) - y__2(t) + Derivative(y__1(t), t),0),Eq(-y__1(t) + y__2(t) + 2*y__3(t) + Derivative(y__2(t), t),0),Eq(y__1(t) + 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)
 

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