76.26.9 problem 9

Internal problem ID [17841]
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 : 9
Date solved : Tuesday, January 28, 2025 at 11:03:47 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x_{1} \left (t \right )&=\frac {3 x_{1} \left (t \right )}{4}+\frac {29 x_{2} \left (t \right )}{4}-\frac {11 x_{3} \left (t \right )}{2}\\ \frac {d}{d t}x_{2} \left (t \right )&=-\frac {3 x_{1} \left (t \right )}{4}+\frac {3 x_{2} \left (t \right )}{4}-\frac {5 x_{3} \left (t \right )}{2}\\ \frac {d}{d t}x_{3} \left (t \right )&=\frac {5 x_{1} \left (t \right )}{4}+\frac {11 x_{2} \left (t \right )}{4}-\frac {5 x_{3} \left (t \right )}{2} \end{align*}

✓ Solution by Maple

Time used: 0.125 (sec). Leaf size: 101

dsolve([diff(x__1(t),t)=3/4*x__1(t)+29/4*x__2(t)-11/2*x__3(t),diff(x__2(t),t)=-3/4*x__1(t)+3/4*x__2(t)-5/2*x__3(t),diff(x__3(t),t)=5/4*x__1(t)+11/4*x__2(t)-5/2*x__3(t)],singsol=all)
 

\begin{align*} x_{1} \left (t \right ) &= {\mathrm e}^{-t} c_{1} +\sin \left (4 t \right ) c_{2} +c_{3} \cos \left (4 t \right ) \\ x_{2} \left (t \right ) &= -{\mathrm e}^{-t} c_{1} +\frac {\sin \left (4 t \right ) c_{2}}{5}+\frac {c_{3} \cos \left (4 t \right )}{5}+\frac {2 c_{2} \cos \left (4 t \right )}{5}-\frac {2 c_{3} \sin \left (4 t \right )}{5} \\ x_{3} \left (t \right ) &= -{\mathrm e}^{-t} c_{1} -\frac {c_{2} \cos \left (4 t \right )}{5}+\frac {c_{3} \sin \left (4 t \right )}{5}+\frac {2 \sin \left (4 t \right ) c_{2}}{5}+\frac {2 c_{3} \cos \left (4 t \right )}{5} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.010 (sec). Leaf size: 186

DSolve[{D[x1[t],t]==3/4*x1[t]+29/4*x2[t]-11/2*x3[t],D[x2[t],t]==-3/4*x1[t]+3/4*x2[t]-5/2*x3[t],D[x3[t],t]==5/4*x1[t]+11/4*x2[t]-5/2*x3[t]},{x1[t],x2[t],x3[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} \text {x1}(t)\to \frac {1}{4} e^{-t} \left ((3 c_1+c_2+2 c_3) e^t \cos (4 t)+(c_1+7 c_2-6 c_3) e^t \sin (4 t)+c_1-c_2-2 c_3\right ) \\ \text {x2}(t)\to \frac {1}{4} e^{-t} \left ((c_1+3 c_2-2 c_3) e^t \cos (4 t)-(c_1-c_2+2 c_3) e^t \sin (4 t)-c_1+c_2+2 c_3\right ) \\ \text {x3}(t)\to \frac {1}{4} e^{-t} \left ((c_1-c_2+2 c_3) e^t \cos (4 t)+(c_1+3 c_2-2 c_3) e^t \sin (4 t)-c_1+c_2+2 c_3\right ) \\ \end{align*}