28.4.51 problem 7.51

Internal problem ID [4583]
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 : 7.51
Date solved : Tuesday, March 04, 2025 at 06:55:57 PM
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 )+4 \,{\mathrm e}^{t}\\ \frac {d}{d t}x_{2} \left (t \right )&=x_{1} \left (t \right )+x_{2} \left (t \right )\\ \frac {d}{d t}x_{3} \left (t \right )&=3 x_{1} \left (t \right )+x_{3} \left (t \right ) \end{align*}

✓ Maple. Time used: 0.160 (sec). Leaf size: 74

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

\begin{align*} x_{1} \left (t \right ) &= {\mathrm e}^{t} \left (c_3 \cos \left (2 t \right )+c_{2} \sin \left (2 t \right )\right ) \\ x_{2} \left (t \right ) &= \frac {{\mathrm e}^{t} \left (c_3 \sin \left (2 t \right )-c_{2} \cos \left (2 t \right )+2 c_{1} -c_{2} \right )}{2} \\ x_{3} \left (t \right ) &= \frac {{\mathrm e}^{t} \left (8+3 c_3 \sin \left (2 t \right )-3 c_{2} \cos \left (2 t \right )-2 c_{1} +c_{2} \right )}{2} \\ \end{align*}

✓ Mathematica. Time used: 0.071 (sec). Leaf size: 158

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

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

✓ Sympy. Time used: 0.191 (sec). Leaf size: 133

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) - 4*exp(t) + Derivative(x__1(t), t),0),Eq(-x__1(t) - x__2(t) + Derivative(x__2(t), t),0),Eq(-3*x__1(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)
 

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