76.27.13 problem 13

Internal problem ID [17780]
Book : Differential equations. An introduction to modern methods and applications. James Brannan, William E. Boyce. Third edition. Wiley 2015
Section : Chapter 6. Systems of First Order Linear Equations. Section 6.5 (Fundamental Matrices and the Exponential of a Matrix). Problems at page 430
Problem number : 13
Date solved : Thursday, March 13, 2025 at 10:49:10 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x_{1} \left (t \right )&=x_{1} \left (t \right )+x_{2} \left (t \right )+x_{3} \left (t \right )\\ \frac {d}{d t}x_{2} \left (t \right )&=2 x_{1} \left (t \right )+x_{2} \left (t \right )-x_{3} \left (t \right )\\ \frac {d}{d t}x_{3} \left (t \right )&=-8 x_{1} \left (t \right )-5 x_{2} \left (t \right )-3 x_{3} \left (t \right ) \end{align*}

✓ Maple. Time used: 0.007 (sec). Leaf size: 66

ode:=[diff(x__1(t),t) = x__1(t)+x__2(t)+x__3(t), diff(x__2(t),t) = 2*x__1(t)+x__2(t)-x__3(t), diff(x__3(t),t) = -8*x__1(t)-5*x__2(t)-3*x__3(t)]; 
dsolve(ode);
 

\begin{align*} x_{1} \left (t \right ) &= c_{2} {\mathrm e}^{-t}+c_{3} {\mathrm e}^{-2 t} \\ x_{2} \left (t \right ) &= c_{1} {\mathrm e}^{2 t}-\frac {4 c_{2} {\mathrm e}^{-t}}{3}-\frac {5 c_{3} {\mathrm e}^{-2 t}}{4} \\ x_{3} \left (t \right ) &= -\frac {2 c_{2} {\mathrm e}^{-t}}{3}-\frac {7 c_{3} {\mathrm e}^{-2 t}}{4}-c_{1} {\mathrm e}^{2 t} \\ \end{align*}

✓ Mathematica. Time used: 0.004 (sec). Leaf size: 160

ode={D[x1[t],t]==1*x1[t]+1*x2[t]+1*x3[t],D[x2[t],t]==2*x1[t]+1*x2[t]-1*x3[t],D[x3[t],t]==-8*x1[t]-5*x2[t]-3*x3[t]}; 
ic={}; 
DSolve[{ode,ic},{x1[t],x2[t],x3[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} \text {x1}(t)\to e^{-2 t} \left (c_1 \left (3 e^t-2\right )+(c_2+c_3) \left (e^t-1\right )\right ) \\ \text {x2}(t)\to \frac {1}{12} e^{-2 t} \left (6 c_1 \left (-8 e^t+3 e^{4 t}+5\right )+c_2 \left (-16 e^t+13 e^{4 t}+15\right )+c_3 \left (-16 e^t+e^{4 t}+15\right )\right ) \\ \text {x3}(t)\to \frac {1}{12} e^{-2 t} \left (-6 c_1 \left (4 e^t+3 e^{4 t}-7\right )+c_2 \left (-8 e^t-13 e^{4 t}+21\right )-c_3 \left (8 e^t+e^{4 t}-21\right )\right ) \\ \end{align*}

✓ Sympy. Time used: 0.142 (sec). Leaf size: 68

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) - x__2(t) - x__3(t) + Derivative(x__1(t), t),0),Eq(-2*x__1(t) - x__2(t) + x__3(t) + Derivative(x__2(t), t),0),Eq(8*x__1(t) + 5*x__2(t) + 3*x__3(t) + Derivative(x__3(t), t),0)] 
ics = {} 
dsolve(ode,func=[x__1(t),x__2(t),x__3(t)],ics=ics)
 

\[ \left [ x^{1}{\left (t \right )} = - \frac {4 C_{1} e^{- 2 t}}{7} - \frac {3 C_{2} e^{- t}}{2}, \ x^{2}{\left (t \right )} = \frac {5 C_{1} e^{- 2 t}}{7} + 2 C_{2} e^{- t} - C_{3} e^{2 t}, \ x^{3}{\left (t \right )} = C_{1} e^{- 2 t} + C_{2} e^{- t} + C_{3} e^{2 t}\right ] \]