7.23.13 problem 19
Internal
problem
ID
[599]
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
5.2
(Applications).
Problems
at
page
345
Problem
number
:
19
Date
solved
:
Wednesday, February 05, 2025 at 01:53:41 AM
CAS
classification
:
system_of_ODEs
\begin{align*} \frac {d}{d t}x \left (t \right )&=x \left (t \right )-2 y \left (t \right )\\ \frac {d}{d t}y \left (t \right )&=-4 x \left (t \right )+4 y \left (t \right )-2 z \left (t \right )\\ \frac {d}{d t}z \left (t \right )&=-4 y \left (t \right )+4 z \left (t \right ) \end{align*}
✓ Solution by Maple
Time used: 0.112 (sec). Leaf size: 912
dsolve([diff(x(t),t)=x(t)-2*y(t),diff(y(t),t)=-4*x(t)+4*y(t)-2*z(t),diff(z(t),t)=-4*y(t)+4*z(t)],singsol=all)
\begin{align*}
x &= \frac {\left (i \sqrt {3}\, \left (81+6 i \sqrt {4962}\right )^{{4}/{3}}-\left (81+6 i \sqrt {4962}\right )^{{4}/{3}}-342 i \left (81+6 i \sqrt {4962}\right )^{{1}/{3}} \sqrt {3}+36 i \sqrt {4962}-2763 i \sqrt {3}-42 \left (81+6 i \sqrt {4962}\right )^{{2}/{3}}+342 \left (81+6 i \sqrt {4962}\right )^{{1}/{3}}-108 \sqrt {1654}-2763\right ) c_1 \,{\mathrm e}^{-\frac {\left (i \left (81+6 i \sqrt {4962}\right )^{{2}/{3}} \sqrt {3}-57 i \sqrt {3}+\left (81+6 i \sqrt {4962}\right )^{{2}/{3}}-18 \left (81+6 i \sqrt {4962}\right )^{{1}/{3}}+57\right ) t}{6 \left (81+6 i \sqrt {4962}\right )^{{1}/{3}}}}}{216 \left (81+6 i \sqrt {4962}\right )^{{2}/{3}}}+\frac {\left (-i \sqrt {3}\, \left (81+6 i \sqrt {4962}\right )^{{4}/{3}}-\left (81+6 i \sqrt {4962}\right )^{{4}/{3}}+342 i \left (81+6 i \sqrt {4962}\right )^{{1}/{3}} \sqrt {3}+36 i \sqrt {4962}+2763 i \sqrt {3}-42 \left (81+6 i \sqrt {4962}\right )^{{2}/{3}}+342 \left (81+6 i \sqrt {4962}\right )^{{1}/{3}}+108 \sqrt {1654}-2763\right ) c_2 \,{\mathrm e}^{\frac {\left (i \left (81+6 i \sqrt {4962}\right )^{{2}/{3}} \sqrt {3}-57 i \sqrt {3}-\left (81+6 i \sqrt {4962}\right )^{{2}/{3}}+18 \left (81+6 i \sqrt {4962}\right )^{{1}/{3}}-57\right ) t}{6 \left (81+6 i \sqrt {4962}\right )^{{1}/{3}}}}}{216 \left (81+6 i \sqrt {4962}\right )^{{2}/{3}}}-\frac {\left (-\left (81+6 i \sqrt {4962}\right )^{{4}/{3}}+36 i \sqrt {4962}+21 \left (81+6 i \sqrt {4962}\right )^{{2}/{3}}+342 \left (81+6 i \sqrt {4962}\right )^{{1}/{3}}-2763\right ) c_3 \,{\mathrm e}^{\frac {\left (\left (81+6 i \sqrt {4962}\right )^{{2}/{3}}+9 \left (81+6 i \sqrt {4962}\right )^{{1}/{3}}+57\right ) t}{3 \left (81+6 i \sqrt {4962}\right )^{{1}/{3}}}}}{108 \left (81+6 i \sqrt {4962}\right )^{{2}/{3}}} \\
y &= c_1 \,{\mathrm e}^{-\frac {\left (i \left (81+6 i \sqrt {4962}\right )^{{2}/{3}} \sqrt {3}-57 i \sqrt {3}+\left (81+6 i \sqrt {4962}\right )^{{2}/{3}}-18 \left (81+6 i \sqrt {4962}\right )^{{1}/{3}}+57\right ) t}{6 \left (81+6 i \sqrt {4962}\right )^{{1}/{3}}}}+c_2 \,{\mathrm e}^{\frac {\left (i \left (81+6 i \sqrt {4962}\right )^{{2}/{3}} \sqrt {3}-57 i \sqrt {3}-\left (81+6 i \sqrt {4962}\right )^{{2}/{3}}+18 \left (81+6 i \sqrt {4962}\right )^{{1}/{3}}-57\right ) t}{6 \left (81+6 i \sqrt {4962}\right )^{{1}/{3}}}}+c_3 \,{\mathrm e}^{\frac {\left (\left (81+6 i \sqrt {4962}\right )^{{2}/{3}}+9 \left (81+6 i \sqrt {4962}\right )^{{1}/{3}}+57\right ) t}{3 \left (81+6 i \sqrt {4962}\right )^{{1}/{3}}}} \\
z &= \frac {\left (-i \sqrt {3}\, \left (81+6 i \sqrt {4962}\right )^{{4}/{3}}+\left (81+6 i \sqrt {4962}\right )^{{4}/{3}}-171 i \left (81+6 i \sqrt {4962}\right )^{{1}/{3}} \sqrt {3}+18 i \sqrt {4962}+3492 i \sqrt {3}+96 \left (81+6 i \sqrt {4962}\right )^{{2}/{3}}+171 \left (81+6 i \sqrt {4962}\right )^{{1}/{3}}-54 \sqrt {1654}+3492\right ) c_1 \,{\mathrm e}^{-\frac {\left (i \left (81+6 i \sqrt {4962}\right )^{{2}/{3}} \sqrt {3}-57 i \sqrt {3}+\left (81+6 i \sqrt {4962}\right )^{{2}/{3}}-18 \left (81+6 i \sqrt {4962}\right )^{{1}/{3}}+57\right ) t}{6 \left (81+6 i \sqrt {4962}\right )^{{1}/{3}}}}}{108 \left (81+6 i \sqrt {4962}\right )^{{2}/{3}}}+\frac {\left (i \sqrt {3}\, \left (81+6 i \sqrt {4962}\right )^{{4}/{3}}+\left (81+6 i \sqrt {4962}\right )^{{4}/{3}}+171 i \left (81+6 i \sqrt {4962}\right )^{{1}/{3}} \sqrt {3}+18 i \sqrt {4962}-3492 i \sqrt {3}+96 \left (81+6 i \sqrt {4962}\right )^{{2}/{3}}+171 \left (81+6 i \sqrt {4962}\right )^{{1}/{3}}+54 \sqrt {1654}+3492\right ) c_2 \,{\mathrm e}^{\frac {\left (i \left (81+6 i \sqrt {4962}\right )^{{2}/{3}} \sqrt {3}-57 i \sqrt {3}-\left (81+6 i \sqrt {4962}\right )^{{2}/{3}}+18 \left (81+6 i \sqrt {4962}\right )^{{1}/{3}}-57\right ) t}{6 \left (81+6 i \sqrt {4962}\right )^{{1}/{3}}}}}{108 \left (81+6 i \sqrt {4962}\right )^{{2}/{3}}}-\frac {\left (\left (81+6 i \sqrt {4962}\right )^{{4}/{3}}+18 i \sqrt {4962}-48 \left (81+6 i \sqrt {4962}\right )^{{2}/{3}}+171 \left (81+6 i \sqrt {4962}\right )^{{1}/{3}}+3492\right ) c_3 \,{\mathrm e}^{\frac {\left (\left (81+6 i \sqrt {4962}\right )^{{2}/{3}}+9 \left (81+6 i \sqrt {4962}\right )^{{1}/{3}}+57\right ) t}{3 \left (81+6 i \sqrt {4962}\right )^{{1}/{3}}}}}{54 \left (81+6 i \sqrt {4962}\right )^{{2}/{3}}} \\
\end{align*}
✓ Solution by Mathematica
Time used: 0.019 (sec). Leaf size: 527
DSolve[{D[x[t],t]==x[t]-2*y[t],D[y[t],t]==-4*x[t]+4*y[t]-2*z[t],D[z[t],t]==-4*y[t]+4*z[t]},{x[t],y[t],z[t]},t,IncludeSingularSolutions -> True]
\begin{align*}
x(t)\to 4 c_3 \text {RootSum}\left [\text {$\#$1}^3-9 \text {$\#$1}^2+8 \text {$\#$1}+24\&,\frac {e^{\text {$\#$1} t}}{3 \text {$\#$1}^2-18 \text {$\#$1}+8}\&\right ]-2 c_2 \text {RootSum}\left [\text {$\#$1}^3-9 \text {$\#$1}^2+8 \text {$\#$1}+24\&,\frac {\text {$\#$1} e^{\text {$\#$1} t}-4 e^{\text {$\#$1} t}}{3 \text {$\#$1}^2-18 \text {$\#$1}+8}\&\right ]+c_1 \text {RootSum}\left [\text {$\#$1}^3-9 \text {$\#$1}^2+8 \text {$\#$1}+24\&,\frac {\text {$\#$1}^2 e^{\text {$\#$1} t}-8 \text {$\#$1} e^{\text {$\#$1} t}+8 e^{\text {$\#$1} t}}{3 \text {$\#$1}^2-18 \text {$\#$1}+8}\&\right ] \\
y(t)\to -4 c_1 \text {RootSum}\left [\text {$\#$1}^3-9 \text {$\#$1}^2+8 \text {$\#$1}+24\&,\frac {\text {$\#$1} e^{\text {$\#$1} t}-4 e^{\text {$\#$1} t}}{3 \text {$\#$1}^2-18 \text {$\#$1}+8}\&\right ]-2 c_3 \text {RootSum}\left [\text {$\#$1}^3-9 \text {$\#$1}^2+8 \text {$\#$1}+24\&,\frac {\text {$\#$1} e^{\text {$\#$1} t}-e^{\text {$\#$1} t}}{3 \text {$\#$1}^2-18 \text {$\#$1}+8}\&\right ]+c_2 \text {RootSum}\left [\text {$\#$1}^3-9 \text {$\#$1}^2+8 \text {$\#$1}+24\&,\frac {\text {$\#$1}^2 e^{\text {$\#$1} t}-5 \text {$\#$1} e^{\text {$\#$1} t}+4 e^{\text {$\#$1} t}}{3 \text {$\#$1}^2-18 \text {$\#$1}+8}\&\right ] \\
z(t)\to 16 c_1 \text {RootSum}\left [\text {$\#$1}^3-9 \text {$\#$1}^2+8 \text {$\#$1}+24\&,\frac {e^{\text {$\#$1} t}}{3 \text {$\#$1}^2-18 \text {$\#$1}+8}\&\right ]-4 c_2 \text {RootSum}\left [\text {$\#$1}^3-9 \text {$\#$1}^2+8 \text {$\#$1}+24\&,\frac {\text {$\#$1} e^{\text {$\#$1} t}-e^{\text {$\#$1} t}}{3 \text {$\#$1}^2-18 \text {$\#$1}+8}\&\right ]+c_3 \text {RootSum}\left [\text {$\#$1}^3-9 \text {$\#$1}^2+8 \text {$\#$1}+24\&,\frac {\text {$\#$1}^2 e^{\text {$\#$1} t}-5 \text {$\#$1} e^{\text {$\#$1} t}-4 e^{\text {$\#$1} t}}{3 \text {$\#$1}^2-18 \text {$\#$1}+8}\&\right ] \\
\end{align*}