28.4.35 problem 7.35

Internal problem ID [4567]
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 : 7.35
Date solved : Tuesday, March 04, 2025 at 06:52:32 PM
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 )\\ \frac {d}{d t}x_{2} \left (t \right )&=x_{1} \left (t \right )+3 x_{2} \left (t \right )-x_{3} \left (t \right )\\ \frac {d}{d t}x_{3} \left (t \right )&=-x_{1} \left (t \right )+2 x_{2} \left (t \right )+3 x_{3} \left (t \right ) \end{align*}

✓ Maple. Time used: 0.060 (sec). Leaf size: 98

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

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

✓ Mathematica. Time used: 0.012 (sec). Leaf size: 139

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

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

✓ Sympy. Time used: 0.204 (sec). Leaf size: 109

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) + Derivative(x__1(t), t),0),Eq(-x__1(t) - 3*x__2(t) + x__3(t) + Derivative(x__2(t), t),0),Eq(x__1(t) - 2*x__2(t) - 3*x__3(t) + Derivative(x__3(t), t),0)] 
ics = {} 
dsolve(ode,func=[x__1(t),x__2(t),x__3(t)],ics=ics)
 

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