7.24.18 problem 28 and 37

Internal problem ID [618]
Book : Elementary Differential Equations. By C. Henry Edwards, David E. Penney and David Calvis. 6th edition. 2008
Section : Chapter 5. Linear systems of differential equations. Section 5.3 (Matrices and linear systems). Problems at page 364
Problem number : 28 and 37
Date solved : Saturday, March 29, 2025 at 05:00:50 PM
CAS classification : system_of_ODEs

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

With initial conditions

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

✓ Maple. Time used: 0.132 (sec). Leaf size: 52

ode:=[diff(x__1(t),t) = x__1(t)+2*x__2(t)+x__3(t), diff(x__2(t),t) = 6*x__1(t)-x__2(t), diff(x__3(t),t) = -x__1(t)-2*x__2(t)-x__3(t)]; 
ic:=x__1(0) = 1x__2(0) = 2x__3(0) = 3; 
dsolve([ode,ic]);
 

\begin{align*} x_{1} \left (t \right ) &= \frac {40 \,{\mathrm e}^{3 t}}{21}-\frac {4 \,{\mathrm e}^{-4 t}}{7}-\frac {1}{3} \\ x_{2} \left (t \right ) &= -2+\frac {20 \,{\mathrm e}^{3 t}}{7}+\frac {8 \,{\mathrm e}^{-4 t}}{7} \\ x_{3} \left (t \right ) &= -\frac {40 \,{\mathrm e}^{3 t}}{21}+\frac {4 \,{\mathrm e}^{-4 t}}{7}+\frac {13}{3} \\ \end{align*}

✓ Mathematica. Time used: 0.007 (sec). Leaf size: 71

ode={D[x1[t],t]==x1[t]+2*x2[t]+x3[t],D[x2[t],t]==6*x1[t]-x2[t],D[x3[t],t]==-x1[t]-2*x2[t]-x3[t]}; 
ic={x1[0]==1,x2[0]==2,x3[0]==3}; 
DSolve[{ode,ic},{x1[t],x2[t],x3[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} \text {x1}(t)\to \frac {1}{21} \left (-12 e^{-4 t}+40 e^{3 t}-7\right ) \\ \text {x2}(t)\to \frac {8 e^{-4 t}}{7}+\frac {20 e^{3 t}}{7}-2 \\ \text {x3}(t)\to \frac {1}{21} \left (12 e^{-4 t}-40 e^{3 t}+91\right ) \\ \end{align*}

✓ Sympy. Time used: 0.183 (sec). Leaf size: 63

from sympy import * 
t = symbols("t") 
x__1 = Function("x__1") 
x__2 = Function("x__2") 
x__3 = Function("x__3") 
ode=[Eq(-x__1(t) - 2*x__2(t) - x__3(t) + Derivative(x__1(t), t),0),Eq(-6*x__1(t) + x__2(t) + Derivative(x__2(t), t),0),Eq(x__1(t) + 2*x__2(t) + x__3(t) + Derivative(x__3(t), t),0)] 
ics = {} 
dsolve(ode,func=[x__1(t),x__2(t),x__3(t)],ics=ics)
 

\[ \left [ x^{1}{\left (t \right )} = - \frac {C_{1}}{13} - C_{2} e^{- 4 t} - C_{3} e^{3 t}, \ x^{2}{\left (t \right )} = - \frac {6 C_{1}}{13} + 2 C_{2} e^{- 4 t} - \frac {3 C_{3} e^{3 t}}{2}, \ x^{3}{\left (t \right )} = C_{1} + C_{2} e^{- 4 t} + C_{3} e^{3 t}\right ] \]