10.16.8 problem 8
Internal
problem
ID
[1408]
Book
:
Elementary
differential
equations
and
boundary
value
problems,
10th
ed.,
Boyce
and
DiPrima
Section
:
Chapter
7.6,
Complex
Eigenvalues.
page
417
Problem
number
:
8
Date
solved
:
Monday, January 27, 2025 at 04:56:53 AM
CAS
classification
:
system_of_ODEs
\begin{align*} \frac {d}{d t}x_{1} \left (t \right )&=-3 x_{1} \left (t \right )+2 x_{3} \left (t \right )\\ \frac {d}{d t}x_{2} \left (t \right )&=x_{1} \left (t \right )-x_{2} \left (t \right )\\ \frac {d}{d t}x_{3} \left (t \right )&=-2 x_{1} \left (t \right )-x_{2} \left (t \right ) \end{align*}
✓ Solution by Maple
Time used: 0.036 (sec). Leaf size: 145
dsolve([diff(x__1(t),t)=-3*x__1(t)+0*x__2(t)+2*x__3(t),diff(x__2(t),t)=1*x__1(t)-1*x__2(t)-0*x__3(t),diff(x__3(t),t)=-2*x__1(t)-1*x__2(t)+0*x__3(t)],singsol=all)
\begin{align*}
x_{1} \left (t \right ) &= c_1 \,{\mathrm e}^{-2 t}+c_2 \,{\mathrm e}^{-t} \sin \left (\sqrt {2}\, t \right )+c_3 \,{\mathrm e}^{-t} \cos \left (\sqrt {2}\, t \right ) \\
x_{2} \left (t \right ) &= -c_1 \,{\mathrm e}^{-2 t}-\frac {c_2 \,{\mathrm e}^{-t} \sqrt {2}\, \cos \left (\sqrt {2}\, t \right )}{2}+\frac {c_3 \,{\mathrm e}^{-t} \sqrt {2}\, \sin \left (\sqrt {2}\, t \right )}{2} \\
x_{3} \left (t \right ) &= \frac {c_1 \,{\mathrm e}^{-2 t}}{2}+c_2 \,{\mathrm e}^{-t} \sin \left (\sqrt {2}\, t \right )+\frac {c_2 \,{\mathrm e}^{-t} \sqrt {2}\, \cos \left (\sqrt {2}\, t \right )}{2}+c_3 \,{\mathrm e}^{-t} \cos \left (\sqrt {2}\, t \right )-\frac {c_3 \,{\mathrm e}^{-t} \sqrt {2}\, \sin \left (\sqrt {2}\, t \right )}{2} \\
\end{align*}
✓ Solution by Mathematica
Time used: 0.033 (sec). Leaf size: 235
DSolve[{D[ x1[t],t]==-3*x1[t]+0*x2[t]+2*x3[t],D[ x2[t],t]==1*x1[t]-1*x2[t]-0*x3[t],D[ x3[t],t]==-2*x1[t]-1*x2[t]+0*x3[t]},{x1[t],x2[t],x3[t]},t,IncludeSingularSolutions -> True]
\begin{align*}
\text {x1}(t)\to \frac {1}{3} e^{-2 t} \left ((c_1+2 (c_2+c_3)) e^t \cos \left (\sqrt {2} t\right )-\sqrt {2} (2 c_1+c_2-2 c_3) e^t \sin \left (\sqrt {2} t\right )+2 (c_1-c_2-c_3)\right ) \\
\text {x2}(t)\to \frac {1}{6} e^{-2 t} \left (2 (2 c_1+c_2-2 c_3) e^t \cos \left (\sqrt {2} t\right )+\sqrt {2} (c_1+2 (c_2+c_3)) e^t \sin \left (\sqrt {2} t\right )+4 (-c_1+c_2+c_3)\right ) \\
\text {x3}(t)\to \frac {1}{6} e^{-2 t} \left (-2 (c_1-c_2-4 c_3) e^t \cos \left (\sqrt {2} t\right )-\sqrt {2} (5 c_1+4 c_2-2 c_3) e^t \sin \left (\sqrt {2} t\right )+2 (c_1-c_2-c_3)\right ) \\
\end{align*}