76.26.2 problem 2

Internal problem ID [17834]
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 : 2
Date solved : Tuesday, January 28, 2025 at 11:03:40 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x_{1} \left (t \right )&=2 x_{1} \left (t \right )-4 x_{2} \left (t \right )-x_{3} \left (t \right )\\ \frac {d}{d t}x_{2} \left (t \right )&=x_{1} \left (t \right )+x_{2} \left (t \right )+3 x_{3} \left (t \right )\\ \frac {d}{d t}x_{3} \left (t \right )&=3 x_{1} \left (t \right )-4 x_{2} \left (t \right )-2 x_{3} \left (t \right ) \end{align*}

✓ Solution by Maple

Time used: 0.136 (sec). Leaf size: 81

dsolve([diff(x__1(t),t)=2*x__1(t)-4*x__2(t)-1*x__3(t),diff(x__2(t),t)=1*x__1(t)+1*x__2(t)+3*x__3(t),diff(x__3(t),t)=3*x__1(t)-4*x__2(t)-2*x__3(t)],singsol=all)
 

\begin{align*} x_{1} \left (t \right ) &= {\mathrm e}^{-t} c_{1} +c_{2} {\mathrm e}^{t} \sin \left (4 t \right )+c_{3} {\mathrm e}^{t} \cos \left (4 t \right ) \\ x_{2} \left (t \right ) &= {\mathrm e}^{-t} c_{1} -c_{2} {\mathrm e}^{t} \cos \left (4 t \right )+c_{3} {\mathrm e}^{t} \sin \left (4 t \right ) \\ x_{3} \left (t \right ) &= -{\mathrm e}^{-t} c_{1} +c_{2} {\mathrm e}^{t} \sin \left (4 t \right )+c_{3} {\mathrm e}^{t} \cos \left (4 t \right ) \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.009 (sec). Leaf size: 168

DSolve[{D[x1[t],t]==2*x1[t]-4*x2[t]-1*x3[t],D[x2[t],t]==1*x1[t]+1*x2[t]+3*x3[t],D[x3[t],t]==3*x1[t]-4*x2[t]-2*x3[t]},{x1[t],x2[t],x3[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} \text {x1}(t)\to \frac {1}{2} e^{-t} \left ((c_1+c_3) e^{2 t} \cos (4 t)+(c_1-2 c_2-c_3) e^{2 t} \sin (4 t)+c_1-c_3\right ) \\ \text {x2}(t)\to \frac {1}{2} e^{-t} \left (-(c_1-2 c_2-c_3) e^{2 t} \cos (4 t)+(c_1+c_3) e^{2 t} \sin (4 t)+c_1-c_3\right ) \\ \text {x3}(t)\to \frac {1}{2} e^{-t} \left ((c_1+c_3) e^{2 t} \cos (4 t)+(c_1-2 c_2-c_3) e^{2 t} \sin (4 t)-c_1+c_3\right ) \\ \end{align*}