8.23.4 problem 4

Internal problem ID [2743]
Book : Differential equations and their applications, 4th ed., M. Braun
Section : Section 3.9, Systems of differential equations. Complex roots. Page 344
Problem number : 4
Date solved : Tuesday, September 30, 2025 at 05:50:28 AM
CAS classification : system_of_ODEs

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

✓ Maple. Time used: 0.226 (sec). Leaf size: 65

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

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

✓ Mathematica. Time used: 0.004 (sec). Leaf size: 67

ode={D[ x1[t],t]==1*x1[t]-0*x2[t]+1*x3[t],D[ x2[t],t]==0*x1[t]+1*x2[t]-1*x3[t],D[ x3[t],t]==-2*x1[t]-0*x2[t]-1*x3[t]}; 
ic={}; 
DSolve[{ode,ic},{x1[t],x2[t],x3[t]},t,IncludeSingularSolutions->True]
 

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

✓ Sympy. Time used: 0.074 (sec). Leaf size: 61

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__3(t) + Derivative(x__1(t), t),0),Eq(-x__2(t) + x__3(t) + Derivative(x__2(t), t),0),Eq(2*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)
 

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