28.4.36 problem 7.36

Internal problem ID [4568]
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 : 7.36
Date solved : Tuesday, March 04, 2025 at 06:52:33 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x_{1} \left (t \right )&=3 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 )&=3 x_{1} \left (t \right )-4 x_{2} \left (t \right )-3 x_{3} \left (t \right )\\ \frac {d}{d t}x_{3} \left (t \right )&=2 x_{1} \left (t \right )-4 x_{2} \left (t \right ) \end{align*}

✓ Maple. Time used: 0.079 (sec). Leaf size: 66

ode:=[diff(x__1(t),t) = 3*x__1(t)-2*x__2(t)-x__3(t), diff(x__2(t),t) = 3*x__1(t)-4*x__2(t)-3*x__3(t), diff(x__3(t),t) = 2*x__1(t)-4*x__2(t)]; 
dsolve(ode);
 

\begin{align*} x_{1} \left (t \right ) &= \frac {c_{2} {\mathrm e}^{-5 t}}{2}+\frac {{\mathrm e}^{2 t} c_3}{2}+{\mathrm e}^{2 t} c_{1} \\ x_{2} \left (t \right ) &= \frac {3 c_{2} {\mathrm e}^{-5 t}}{2}-\frac {{\mathrm e}^{2 t} c_3}{4}+\frac {{\mathrm e}^{2 t} c_{1}}{2} \\ x_{3} \left (t \right ) &= c_{2} {\mathrm e}^{-5 t}+{\mathrm e}^{2 t} c_3 \\ \end{align*}

✓ Mathematica. Time used: 0.01 (sec). Leaf size: 135

ode={D[x1[t],t]==3*x1[t]-2*x2[t]-x3[t],D[x2[t],t]==3*x1[t]-4*x2[t]-3*x3[t],D[x3[t],t]==2*x1[t]-4*x2[t]}; 
ic={}; 
DSolve[{ode,ic},{x1[t],x2[t],x3[t]},t,IncludeSingularSolutions->True]
 

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

✓ Sympy. Time used: 0.140 (sec). Leaf size: 54

from sympy import * 
t = symbols("t") 
x__1 = Function("x__1") 
x__2 = Function("x__2") 
x__3 = Function("x__3") 
ode=[Eq(-3*x__1(t) + 2*x__2(t) + x__3(t) + Derivative(x__1(t), t),0),Eq(-3*x__1(t) + 4*x__2(t) + 3*x__3(t) + Derivative(x__2(t), t),0),Eq(-2*x__1(t) + 4*x__2(t) + Derivative(x__3(t), t),0)] 
ics = {} 
dsolve(ode,func=[x__1(t),x__2(t),x__3(t)],ics=ics)
 

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