76.26.13 problem 17
Internal
problem
ID
[17845]
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
:
17
Date
solved
:
Tuesday, January 28, 2025 at 11:03:56 AM
CAS
classification
:
system_of_ODEs
\begin{align*} \frac {d}{d t}x_{1} \left (t \right )&=-3 x_{1} \left (t \right )-5 x_{2} \left (t \right )+8 x_{3} \left (t \right )+14 x_{4} \left (t \right )\\ \frac {d}{d t}x_{2} \left (t \right )&=-6 x_{1} \left (t \right )-8 x_{2} \left (t \right )+11 x_{3} \left (t \right )+27 x_{4} \left (t \right )\\ \frac {d}{d t}x_{3} \left (t \right )&=-6 x_{1} \left (t \right )-4 x_{2} \left (t \right )+7 x_{3} \left (t \right )+17 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 )+4 x_{4} \left (t \right ) \end{align*}
✓ Solution by Maple
Time used: 2.080 (sec). Leaf size: 145
dsolve([diff(x__1(t),t)=-3*x__1(t)-5*x__2(t)+8*x__3(t)+14*x__4(t),diff(x__2(t),t)=-6*x__1(t)-8*x__2(t)+11*x__3(t)+27*x__4(t),diff(x__3(t),t)=-6*x__1(t)-4*x__2(t)+7*x__3(t)+17*x__4(t),diff(x__4(t),t)=0*x__1(t)-2*x__2(t)+2*x__3(t)+4*x__4(t)],singsol=all)
\begin{align*}
x_{1} \left (t \right ) &= \frac {\cos \left (3 t \right ) c_{1}}{2}-\frac {c_{2} \cos \left (3 t \right )}{2}+\frac {c_{1} \sin \left (3 t \right )}{2}+\frac {\sin \left (3 t \right ) c_{2}}{2}+\cos \left (2 t \right ) c_{3} +c_4 \cos \left (2 t \right )+c_{3} \sin \left (2 t \right )-c_4 \sin \left (2 t \right ) \\
x_{2} \left (t \right ) &= \cos \left (3 t \right ) c_{1} +\sin \left (3 t \right ) c_{2} -\cos \left (2 t \right ) c_{3} +c_4 \cos \left (2 t \right )+c_{3} \sin \left (2 t \right )+c_4 \sin \left (2 t \right ) \\
x_{3} \left (t \right ) &= \cos \left (3 t \right ) c_{1} +\sin \left (3 t \right ) c_{2} -c_4 \cos \left (2 t \right )-c_{3} \sin \left (2 t \right ) \\
x_{4} \left (t \right ) &= c_4 \cos \left (2 t \right )+c_{3} \sin \left (2 t \right ) \\
\end{align*}
✓ Solution by Mathematica
Time used: 0.019 (sec). Leaf size: 274
DSolve[{D[x1[t],t]==-3*x1[t]-5*x2[t]+8*x3[t]+14*x4[t],D[x2[t],t]==-6*x1[t]-8*x2[t]+11*x3[t]+27*x4[t],D[x3[t],t]==-6*x1[t]-4*x2[t]+7*x3[t]+17*x4[t],D[x4[t],t]==0*x1[t]-2*x2[t]+2*x3[t]+4*x4[t]},{x1[t],x2[t],x3[t],x4[t]},t,IncludeSingularSolutions -> True]
\begin{align*}
\text {x1}(t)\to c_1 (\cos (3 t)-\sin (3 t))-c_2 (\sin (2 t)+\sin (3 t)+\cos (2 t)-\cos (3 t))+c_3 (\sin (2 t)+2 \sin (3 t)+\cos (2 t)-\cos (3 t))+c_4 (\sin (2 t)+4 \sin (3 t)+3 \cos (2 t)-3 \cos (3 t)) \\
\text {x2}(t)\to -2 c_1 \sin (3 t)+c_2 (-\sin (2 t)-2 \sin (3 t)+\cos (2 t))+c_3 (\sin (2 t)+3 \sin (3 t)-\cos (2 t)+\cos (3 t))+c_4 (3 \sin (2 t)+7 \sin (3 t)-\cos (2 t)+\cos (3 t)) \\
\text {x3}(t)\to c_2 (\sin (2 t)-2 \sin (3 t))-2 c_1 \sin (3 t)+c_3 (-\sin (2 t)+3 \sin (3 t)+\cos (3 t))+c_4 (-2 \sin (2 t)+7 \sin (3 t)-\cos (2 t)+\cos (3 t)) \\
\text {x4}(t)\to c_4 \cos (2 t)+(-c_2+c_3+2 c_4) \sin (2 t) \\
\end{align*}