7.24.9 problem 19
Internal
problem
ID
[609]
Book
:
Elementary
Differential
Equations.
By
C.
Henry
Edwards,
David
E.
Penney
and
David
Calvis.
6th
edition.
2008
Section
:
Chapter
5.
Linear
systems
of
differential
equations.
Section
5.3
(Matrices
and
linear
systems).
Problems
at
page
364
Problem
number
:
19
Date
solved
:
Wednesday, February 05, 2025 at 03:47:09 AM
CAS
classification
:
system_of_ODEs
\begin{align*} \frac {d}{d t}x_{1} \left (t \right )&=x_{2} \left (t \right )\\ \frac {d}{d t}x_{2} \left (t \right )&=2 x_{3} \left (t \right )\\ \frac {d}{d t}x_{3} \left (t \right )&=3 x_{4} \left (t \right )\\ \frac {d}{d t}x_{4} \left (t \right )&=4 x_{1} \left (t \right ) \end{align*}
✓ Solution by Maple
Time used: 0.052 (sec). Leaf size: 169
dsolve([diff(x__1(t),t)=x__2(t),diff(x__2(t),t)=2*x__3(t),diff(x__3(t),t)=3*x__4(t),diff(x__4(t),t)=4*x__1(t)],singsol=all)
\begin{align*}
x_{1} \left (t \right ) &= c_1 \,{\mathrm e}^{-24^{{1}/{4}} t}+c_2 \,{\mathrm e}^{24^{{1}/{4}} t}-c_3 \sin \left (24^{{1}/{4}} t \right )+c_4 \cos \left (24^{{1}/{4}} t \right ) \\
x_{2} \left (t \right ) &= -24^{{1}/{4}} \left (c_1 \,{\mathrm e}^{-24^{{1}/{4}} t}-c_2 \,{\mathrm e}^{24^{{1}/{4}} t}+\cos \left (24^{{1}/{4}} t \right ) c_3 +\sin \left (24^{{1}/{4}} t \right ) c_4 \right ) \\
x_{3} \left (t \right ) &= \sqrt {6}\, \left (c_1 \,{\mathrm e}^{-24^{{1}/{4}} t}+c_2 \,{\mathrm e}^{24^{{1}/{4}} t}-c_4 \cos \left (24^{{1}/{4}} t \right )+c_3 \sin \left (24^{{1}/{4}} t \right )\right ) \\
x_{4} \left (t \right ) &= -\frac {24^{{3}/{4}} \left (c_1 \,{\mathrm e}^{-24^{{1}/{4}} t}-c_2 \,{\mathrm e}^{24^{{1}/{4}} t}-\cos \left (24^{{1}/{4}} t \right ) c_3 -\sin \left (24^{{1}/{4}} t \right ) c_4 \right )}{6} \\
\end{align*}
✓ Solution by Mathematica
Time used: 0.025 (sec). Leaf size: 400
DSolve[{D[x1[t],t]==x2[t],D[x2[t],t]==2*x3[t],D[x3[t],t]==3*x4[t],D[x4[t],t]==4*x1[t]},{x1[t],x2[t],x3[t],x4[t]},t,IncludeSingularSolutions -> True]
\begin{align*}
\text {x1}(t)\to \frac {1}{4} c_1 \text {RootSum}\left [\text {$\#$1}^4-24\&,e^{\text {$\#$1} t}\&\right ]+\frac {1}{4} c_2 \text {RootSum}\left [\text {$\#$1}^4-24\&,\frac {e^{\text {$\#$1} t}}{\text {$\#$1}}\&\right ]+\frac {3}{2} c_4 \text {RootSum}\left [\text {$\#$1}^4-24\&,\frac {e^{\text {$\#$1} t}}{\text {$\#$1}^3}\&\right ]+\frac {1}{2} c_3 \text {RootSum}\left [\text {$\#$1}^4-24\&,\frac {e^{\text {$\#$1} t}}{\text {$\#$1}^2}\&\right ] \\
\text {x2}(t)\to \frac {1}{4} c_2 \text {RootSum}\left [\text {$\#$1}^4-24\&,e^{\text {$\#$1} t}\&\right ]+\frac {1}{2} c_3 \text {RootSum}\left [\text {$\#$1}^4-24\&,\frac {e^{\text {$\#$1} t}}{\text {$\#$1}}\&\right ]+6 c_1 \text {RootSum}\left [\text {$\#$1}^4-24\&,\frac {e^{\text {$\#$1} t}}{\text {$\#$1}^3}\&\right ]+\frac {3}{2} c_4 \text {RootSum}\left [\text {$\#$1}^4-24\&,\frac {e^{\text {$\#$1} t}}{\text {$\#$1}^2}\&\right ] \\
\text {x3}(t)\to \frac {1}{4} c_3 \text {RootSum}\left [\text {$\#$1}^4-24\&,e^{\text {$\#$1} t}\&\right ]+\frac {3}{4} c_4 \text {RootSum}\left [\text {$\#$1}^4-24\&,\frac {e^{\text {$\#$1} t}}{\text {$\#$1}}\&\right ]+3 c_2 \text {RootSum}\left [\text {$\#$1}^4-24\&,\frac {e^{\text {$\#$1} t}}{\text {$\#$1}^3}\&\right ]+3 c_1 \text {RootSum}\left [\text {$\#$1}^4-24\&,\frac {e^{\text {$\#$1} t}}{\text {$\#$1}^2}\&\right ] \\
\text {x4}(t)\to \frac {1}{4} c_4 \text {RootSum}\left [\text {$\#$1}^4-24\&,e^{\text {$\#$1} t}\&\right ]+c_1 \text {RootSum}\left [\text {$\#$1}^4-24\&,\frac {e^{\text {$\#$1} t}}{\text {$\#$1}}\&\right ]+2 c_3 \text {RootSum}\left [\text {$\#$1}^4-24\&,\frac {e^{\text {$\#$1} t}}{\text {$\#$1}^3}\&\right ]+c_2 \text {RootSum}\left [\text {$\#$1}^4-24\&,\frac {e^{\text {$\#$1} t}}{\text {$\#$1}^2}\&\right ] \\
\end{align*}