76.26.12 problem 16
Internal
problem
ID
[17844]
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
:
16
Date
solved
:
Tuesday, January 28, 2025 at 11:03:54 AM
CAS
classification
:
system_of_ODEs
\begin{align*} \frac {d}{d t}x_{1} \left (t \right )&=-3 x_{1} \left (t \right )-4 x_{2} \left (t \right )+5 x_{3} \left (t \right )+9 x_{4} \left (t \right )\\ \frac {d}{d t}x_{2} \left (t \right )&=-2 x_{1} \left (t \right )-5 x_{2} \left (t \right )+4 x_{3} \left (t \right )+12 x_{4} \left (t \right )\\ \frac {d}{d t}x_{3} \left (t \right )&=-2 x_{1} \left (t \right )-x_{3} \left (t \right )+2 x_{4} \left (t \right )\\ \frac {d}{d t}x_{4} \left (t \right )&=-2 x_{2} \left (t \right )+2 x_{3} \left (t \right )+3 x_{4} \left (t \right ) \end{align*}
✓ Solution by Maple
Time used: 0.163 (sec). Leaf size: 209
dsolve([diff(x__1(t),t)=-3*x__1(t)-4*x__2(t)+5*x__3(t)+9*x__4(t),diff(x__2(t),t)=-2*x__1(t)-5*x__2(t)+4*x__3(t)+12*x__4(t),diff(x__3(t),t)=-2*x__1(t)+0*x__2(t)-1*x__3(t)+2*x__4(t),diff(x__4(t),t)=0*x__1(t)-2*x__2(t)+2*x__3(t)+3*x__4(t)],singsol=all)
\begin{align*}
x_{1} \left (t \right ) &= \frac {{\mathrm e}^{-2 t} \sin \left (t \right ) c_{2}}{2}-\frac {{\mathrm e}^{-2 t} \cos \left (t \right ) c_{1}}{2}-c_{3} {\mathrm e}^{-t} \cos \left (2 t \right )-c_{3} {\mathrm e}^{-t} \sin \left (2 t \right )-c_4 \,{\mathrm e}^{-t} \cos \left (2 t \right )+c_4 \,{\mathrm e}^{-t} \sin \left (2 t \right )+\frac {c_{1} {\mathrm e}^{-2 t} \sin \left (t \right )}{2}+\frac {c_{2} {\mathrm e}^{-2 t} \cos \left (t \right )}{2} \\
x_{2} \left (t \right ) &= c_{3} {\mathrm e}^{-t} \cos \left (2 t \right )-c_{3} {\mathrm e}^{-t} \sin \left (2 t \right )-c_4 \,{\mathrm e}^{-t} \cos \left (2 t \right )-c_4 \,{\mathrm e}^{-t} \sin \left (2 t \right )+c_{1} {\mathrm e}^{-2 t} \sin \left (t \right )+c_{2} {\mathrm e}^{-2 t} \cos \left (t \right ) \\
x_{3} \left (t \right ) &= c_{1} {\mathrm e}^{-2 t} \sin \left (t \right )+c_{2} {\mathrm e}^{-2 t} \cos \left (t \right )+c_{3} {\mathrm e}^{-t} \sin \left (2 t \right )+c_4 \,{\mathrm e}^{-t} \cos \left (2 t \right ) \\
x_{4} \left (t \right ) &= -{\mathrm e}^{-t} \left (c_4 \cos \left (2 t \right )+c_{3} \sin \left (2 t \right )\right ) \\
\end{align*}
✓ Solution by Mathematica
Time used: 0.017 (sec). Leaf size: 258
DSolve[{D[x1[t],t]==-3*x1[t]-4*x2[t]+5*x3[t]+9*x4[t],D[x2[t],t]==-2*x1[t]-5*x2[t]+4*x3[t]+12*x4[t],D[x3[t],t]==-2*x1[t]+0*x2[t]-1*x3[t]+2*x4[t],D[x4[t],t]==0*x1[t]-2*x2[t]+2*x3[t]+3*x4[t]},{x1[t],x2[t],x3[t],x4[t]},t,IncludeSingularSolutions -> True]
\begin{align*}
\text {x1}(t)\to e^{-2 t} \left ((c_1+c_2-c_3-3 c_4) \cos (t)-(c_2-c_3-3 c_4) e^t \cos (2 t)-\sin (t) \left (2 (c_2-c_3-c_4) e^t \cos (t)+c_1+c_2-2 c_3-4 c_4\right )\right ) \\
\text {x2}(t)\to e^{-2 t} \left ((c_2-c_3-c_4) e^t \cos (2 t)+(-2 c_1-2 c_2+3 c_3+7 c_4) \sin (t)+\cos (t) \left (-2 (c_2-c_3-3 c_4) e^t \sin (t)+c_3+c_4\right )\right ) \\
\text {x3}(t)\to e^{-2 t} \left (c_4 \left (-e^t\right ) \cos (2 t)+(-2 c_1-2 c_2+3 c_3+7 c_4) \sin (t)+\cos (t) \left (2 (c_2-c_3-2 c_4) e^t \sin (t)+c_3+c_4\right )\right ) \\
\text {x4}(t)\to e^{-t} (c_4 \cos (2 t)+(-c_2+c_3+2 c_4) \sin (2 t)) \\
\end{align*}