76.26.4 problem 4

Internal problem ID [17836]
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 : 4
Date solved : Tuesday, January 28, 2025 at 11:03:42 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x_{1} \left (t \right )&=-4 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 )&=-6 x_{1} \left (t \right )-3 x_{3} \left (t \right )\\ \frac {d}{d t}x_{3} \left (t \right )&=\frac {8 x_{2} \left (t \right )}{3}-2 x_{3} \left (t \right ) \end{align*}

✓ Solution by Maple

Time used: 0.463 (sec). Leaf size: 96

dsolve([diff(x__1(t),t)=-4*x__1(t)+2*x__2(t)-1*x__3(t),diff(x__2(t),t)=-6*x__1(t)+0*x__2(t)-3*x__3(t),diff(x__3(t),t)=0*x__1(t)+8/3*x__2(t)-2*x__3(t)],singsol=all)
 

\begin{align*} x_{1} \left (t \right ) &= -\frac {{\mathrm e}^{-2 t} \left (2 c_{2} \cos \left (4 t \right )-2 c_{3} \cos \left (4 t \right )-2 \sin \left (4 t \right ) c_{2} -2 c_{3} \sin \left (4 t \right )+3 c_{1} +2 c_{2} \right )}{6} \\ x_{2} \left (t \right ) &= {\mathrm e}^{-2 t} \left (c_{3} \cos \left (4 t \right )+\sin \left (4 t \right ) c_{2} \right ) \\ x_{3} \left (t \right ) &= -\frac {{\mathrm e}^{-2 t} \left (2 c_{2} \cos \left (4 t \right )-2 c_{3} \sin \left (4 t \right )-3 c_{1} -2 c_{2} \right )}{3} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.009 (sec). Leaf size: 152

DSolve[{D[x1[t],t]==-4*x1[t]+2*x2[t]-1*x3[t],D[x2[t],t]==-6*x1[t]+0*x2[t]-3*x3[t],D[x3[t],t]==0*x1[t]+8/3*x2[t]-2*x3[t]},{x1[t],x2[t],x3[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} \text {x1}(t)\to \frac {1}{12} e^{-2 t} ((6 c_1+2 c_2+3 c_3) \cos (4 t)-3 (2 c_1-2 c_2+c_3) \sin (4 t)+6 c_1-2 c_2-3 c_3) \\ \text {x2}(t)\to \frac {1}{4} e^{-2 t} (4 c_2 \cos (4 t)+(-6 c_1+2 c_2-3 c_3) \sin (4 t)) \\ \text {x3}(t)\to \frac {1}{6} e^{-2 t} ((6 c_1-2 c_2+3 c_3) \cos (4 t)+4 c_2 \sin (4 t)-6 c_1+2 c_2+3 c_3) \\ \end{align*}