7.24.6 problem 16
Internal
problem
ID
[606]
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
:
16
Date
solved
:
Wednesday, February 05, 2025 at 03:46:00 AM
CAS
classification
:
system_of_ODEs
\begin{align*} \frac {d}{d t}x \left (t \right )&=2 x \left (t \right )-3 y \left (t \right )\\ \frac {d}{d t}y \left (t \right )&=x \left (t \right )+y \left (t \right )+2 z \left (t \right )\\ \frac {d}{d t}z \left (t \right )&=5 y \left (t \right )-7 z \left (t \right ) \end{align*}
✓ Solution by Maple
Time used: 0.322 (sec). Leaf size: 3112
dsolve([diff(x(t),t)=2*x(t)-3*y(t),diff(y(t),t)=x(t)+y(t)+2*z(t),diff(z(t),t)=5*y(t)-7*z(t)],singsol=all)
\begin{align*}
\text {Expression too large to display} \\
y \left (t \right ) &= \sin \left (\frac {\left (\left (10196+12 \sqrt {352785}\right )^{{2}/{3}}-376\right ) t \sqrt {3}\, 2^{{1}/{3}}}{24 \left (2549+3 \sqrt {352785}\right )^{{1}/{3}}}\right ) {\mathrm e}^{\frac {\left (376+\left (10196+12 \sqrt {352785}\right )^{{2}/{3}}-16 \left (10196+12 \sqrt {352785}\right )^{{1}/{3}}\right ) t}{12 \left (10196+12 \sqrt {352785}\right )^{{1}/{3}}}} c_2 +{\mathrm e}^{\frac {\left (376+\left (10196+12 \sqrt {352785}\right )^{{2}/{3}}-16 \left (10196+12 \sqrt {352785}\right )^{{1}/{3}}\right ) t}{12 \left (10196+12 \sqrt {352785}\right )^{{1}/{3}}}} \cos \left (\frac {\left (\left (10196+12 \sqrt {352785}\right )^{{2}/{3}}-376\right ) t \sqrt {3}\, 2^{{1}/{3}}}{24 \left (2549+3 \sqrt {352785}\right )^{{1}/{3}}}\right ) c_3 +c_1 \,{\mathrm e}^{-\frac {\left (\left (10196+12 \sqrt {352785}\right )^{{2}/{3}}+8 \left (10196+12 \sqrt {352785}\right )^{{1}/{3}}+376\right ) t}{6 \left (10196+12 \sqrt {352785}\right )^{{1}/{3}}}} \\
\text {Expression too large to display} \\
\end{align*}
✓ Solution by Mathematica
Time used: 0.019 (sec). Leaf size: 526
DSolve[{D[x[t],t]==2*x[t]-3*y[t],D[y[t],t]==x[t]+y[t]+2*z[t],D[z[t],t]==5*y[t]-7*z[t]},{x[t],y[t],z[t]},t,IncludeSingularSolutions -> True]
\begin{align*}
x(t)\to -6 c_3 \text {RootSum}\left [\text {$\#$1}^3+4 \text {$\#$1}^2-26 \text {$\#$1}+55\&,\frac {e^{\text {$\#$1} t}}{3 \text {$\#$1}^2+8 \text {$\#$1}-26}\&\right ]-3 c_2 \text {RootSum}\left [\text {$\#$1}^3+4 \text {$\#$1}^2-26 \text {$\#$1}+55\&,\frac {\text {$\#$1} e^{\text {$\#$1} t}+7 e^{\text {$\#$1} t}}{3 \text {$\#$1}^2+8 \text {$\#$1}-26}\&\right ]+c_1 \text {RootSum}\left [\text {$\#$1}^3+4 \text {$\#$1}^2-26 \text {$\#$1}+55\&,\frac {\text {$\#$1}^2 e^{\text {$\#$1} t}+6 \text {$\#$1} e^{\text {$\#$1} t}-17 e^{\text {$\#$1} t}}{3 \text {$\#$1}^2+8 \text {$\#$1}-26}\&\right ] \\
y(t)\to 2 c_3 \text {RootSum}\left [\text {$\#$1}^3+4 \text {$\#$1}^2-26 \text {$\#$1}+55\&,\frac {\text {$\#$1} e^{\text {$\#$1} t}-2 e^{\text {$\#$1} t}}{3 \text {$\#$1}^2+8 \text {$\#$1}-26}\&\right ]+c_1 \text {RootSum}\left [\text {$\#$1}^3+4 \text {$\#$1}^2-26 \text {$\#$1}+55\&,\frac {\text {$\#$1} e^{\text {$\#$1} t}+7 e^{\text {$\#$1} t}}{3 \text {$\#$1}^2+8 \text {$\#$1}-26}\&\right ]+c_2 \text {RootSum}\left [\text {$\#$1}^3+4 \text {$\#$1}^2-26 \text {$\#$1}+55\&,\frac {\text {$\#$1}^2 e^{\text {$\#$1} t}+5 \text {$\#$1} e^{\text {$\#$1} t}-14 e^{\text {$\#$1} t}}{3 \text {$\#$1}^2+8 \text {$\#$1}-26}\&\right ] \\
z(t)\to 5 c_1 \text {RootSum}\left [\text {$\#$1}^3+4 \text {$\#$1}^2-26 \text {$\#$1}+55\&,\frac {e^{\text {$\#$1} t}}{3 \text {$\#$1}^2+8 \text {$\#$1}-26}\&\right ]+5 c_2 \text {RootSum}\left [\text {$\#$1}^3+4 \text {$\#$1}^2-26 \text {$\#$1}+55\&,\frac {\text {$\#$1} e^{\text {$\#$1} t}-2 e^{\text {$\#$1} t}}{3 \text {$\#$1}^2+8 \text {$\#$1}-26}\&\right ]+c_3 \text {RootSum}\left [\text {$\#$1}^3+4 \text {$\#$1}^2-26 \text {$\#$1}+55\&,\frac {\text {$\#$1}^2 e^{\text {$\#$1} t}-3 \text {$\#$1} e^{\text {$\#$1} t}+5 e^{\text {$\#$1} t}}{3 \text {$\#$1}^2+8 \text {$\#$1}-26}\&\right ] \\
\end{align*}