28.4.54 problem 7.54

Internal problem ID [4586]
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 : 7.54
Date solved : Sunday, March 30, 2025 at 03:29:41 AM
CAS classification : system_of_ODEs

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

✓ Maple. Time used: 0.181 (sec). Leaf size: 60

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

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

✓ Mathematica. Time used: 0.033 (sec). Leaf size: 119

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

\begin{align*} \text {x1}(t)\to 3 e^{2 t}+(-6+2 c_1-c_2-c_3) e^t+2-c_1+c_2+c_3 \\ \text {x2}(t)\to 3 e^{2 t}+(-9+3 c_1-2 c_2-3 c_3) e^t+3 (2-c_1+c_2+c_3) \\ \text {x3}(t)\to -e^{2 t}+(3-c_1+c_2+2 c_3) e^t-2+c_1-c_2-c_3 \\ \end{align*}

✓ Sympy. Time used: 0.171 (sec). Leaf size: 48

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

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