28.4.48 problem 7.48

Internal problem ID [4580]
Book : Differential equations for engineers by Wei-Chau XIE, Cambridge Press 2010
Section : Chapter 7. Systems of linear differential equations. Problems at page 351
Problem number : 7.48
Date solved : Sunday, March 30, 2025 at 03:26:47 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 )&=x_{1} \left (t \right )+x_{2} \left (t \right )-x_{3} \left (t \right )+6 \,{\mathrm e}^{-t}\\ \frac {d}{d t}x_{3} \left (t \right )&=2 x_{1} \left (t \right )-x_{2} \left (t \right ) \end{align*}

✓ Maple. Time used: 0.146 (sec). Leaf size: 69

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

\begin{align*} x_{1} \left (t \right ) &= c_1 \,{\mathrm e}^{2 t}+c_2 \,{\mathrm e}^{t}-\frac {c_3 \,{\mathrm e}^{-t}}{5}-\frac {6 \,{\mathrm e}^{-t}}{5} \\ x_{2} \left (t \right ) &= c_2 \,{\mathrm e}^{t}+\frac {3 c_3 \,{\mathrm e}^{-t}}{5}-\frac {12 \,{\mathrm e}^{-t}}{5} \\ x_{3} \left (t \right ) &= c_1 \,{\mathrm e}^{2 t}+c_2 \,{\mathrm e}^{t}+c_3 \,{\mathrm e}^{-t} \\ \end{align*}

✓ Mathematica. Time used: 0.106 (sec). Leaf size: 177

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

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

✓ Sympy. Time used: 0.212 (sec). Leaf size: 60

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(-x__1(t) - x__2(t) + x__3(t) + Derivative(x__2(t), t) - 6*exp(-t),0),Eq(-2*x__1(t) + x__2(t) + Derivative(x__3(t), t),0)] 
ics = {} 
dsolve(ode,func=[x__1(t),x__2(t),x__3(t)],ics=ics)
 

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