22.4.38 problem 7.38

Internal problem ID [4570]
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 : 7.38
Date solved : Tuesday, September 30, 2025 at 07:34:27 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 )-2 x_{3} \left (t \right )\\ \frac {d}{d t}x_{2} \left (t \right )&=4 x_{1} \left (t \right )+x_{2} \left (t \right )\\ \frac {d}{d t}x_{3} \left (t \right )&=2 x_{1} \left (t \right )+x_{2} \left (t \right )-x_{3} \left (t \right ) \end{align*}

✓ Maple. Time used: 0.117 (sec). Leaf size: 73

ode:=[diff(x__1(t),t) = -x__1(t)+x__2(t)-2*x__3(t), diff(x__2(t),t) = 4*x__1(t)+x__2(t), diff(x__3(t),t) = 2*x__1(t)+x__2(t)-x__3(t)]; 
dsolve(ode);
 

\begin{align*} x_{1} \left (t \right ) &= -\frac {{\mathrm e}^{-t} \left (2 c_3 t +2 c_2 -c_3 \right )}{4} \\ x_{2} \left (t \right ) &= c_1 \,{\mathrm e}^{t}+c_2 \,{\mathrm e}^{-t}+c_3 \,{\mathrm e}^{-t} t \\ x_{3} \left (t \right ) &= \frac {c_1 \,{\mathrm e}^{t}}{2}+\frac {c_2 \,{\mathrm e}^{-t}}{2}+\frac {c_3 \,{\mathrm e}^{-t} t}{2}+\frac {c_3 \,{\mathrm e}^{-t}}{4} \\ \end{align*}

✓ Mathematica. Time used: 0.004 (sec). Leaf size: 120

ode={D[x1[t],t]==-x1[t]+x2[t]-2*x3[t],D[x2[t],t]==4*x1[t]+x2[t],D[x3[t],t]==2*x1[t]+x2[t]-x3[t]}; 
ic={}; 
DSolve[{ode,ic},{x1[t],x2[t],x3[t]},t,IncludeSingularSolutions->True]
 

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

✓ Sympy. Time used: 0.081 (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) + 2*x__3(t) + Derivative(x__1(t), t),0),Eq(-4*x__1(t) - x__2(t) + Derivative(x__2(t), t),0),Eq(-2*x__1(t) - 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)
 

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