28.4.34 problem 7.34

Internal problem ID [4566]
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 : 7.34
Date solved : Sunday, March 30, 2025 at 03:26:25 AM
CAS classification : system_of_ODEs

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

✓ Maple. Time used: 0.124 (sec). Leaf size: 65

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

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

✓ Mathematica. Time used: 0.005 (sec). Leaf size: 169

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

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

✓ Sympy. Time used: 0.140 (sec). Leaf size: 70

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) + 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),0),Eq(-4*x__1(t) + x__2(t) - 4*x__3(t) + Derivative(x__3(t), t),0)] 
ics = {} 
dsolve(ode,func=[x__1(t),x__2(t),x__3(t)],ics=ics)
 

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