76.26.1 problem 1

Internal problem ID [17762]
Book : Differential equations. An introduction to modern methods and applications. James Brannan, William E. Boyce. Third edition. Wiley 2015
Section : Chapter 6. Systems of First Order Linear Equations. Section 6.4 (Nondefective Matrices with Complex Eigenvalues). Problems at page 419
Problem number : 1
Date solved : Monday, March 31, 2025 at 04:26:57 PM
CAS classification : system_of_ODEs

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

✓ Maple. Time used: 0.148 (sec). Leaf size: 75

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

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

✓ Mathematica. Time used: 0.009 (sec). Leaf size: 111

ode={D[x1[t],t]==-2*x1[t]+2*x2[t]+1*x3[t],D[x2[t],t]==-2*x1[t]+2*x2[t]+2*x3[t],D[x3[t],t]==2*x1[t]-3*x2[t]-3*x3[t]}; 
ic={}; 
DSolve[{ode,ic},{x1[t],x2[t],x3[t]},t,IncludeSingularSolutions->True]
 

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

✓ Sympy. Time used: 0.140 (sec). Leaf size: 76

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) - 2*x__2(t) - x__3(t) + Derivative(x__1(t), t),0),Eq(2*x__1(t) - 2*x__2(t) - 2*x__3(t) + Derivative(x__2(t), t),0),Eq(-2*x__1(t) + 3*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)
 

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