52.9.4 problem 4
Internal
problem
ID
[8382]
Book
:
DIFFERENTIAL
EQUATIONS
with
Boundary
Value
Problems.
DENNIS
G.
ZILL,
WARREN
S.
WRIGHT,
MICHAEL
R.
CULLEN.
Brooks/Cole.
Boston,
MA.
2013.
8th
edition.
Section
:
CHAPTER
8
SYSTEMS
OF
LINEAR
FIRST-ORDER
DIFFERENTIAL
EQUATIONS.
EXERCISES
8.1.
Page
332
Problem
number
:
4
Date
solved
:
Monday, January 27, 2025 at 03:50:04 PM
CAS
classification
:
system_of_ODEs
\begin{align*} x^{\prime }\left (t \right )&=x \left (t \right )-y\\ y^{\prime }&=x \left (t \right )+2 z \left (t \right )\\ z^{\prime }\left (t \right )&=-x \left (t \right )+z \left (t \right ) \end{align*}
✓ Solution by Maple
Time used: 0.349 (sec). Leaf size: 2265
dsolve([diff(x(t),t)=x(t)-y(t),diff(y(t),t)=x(t)+2*z(t),diff(z(t),t)=-x(t)+z(t)],singsol=all)
\begin{align*}
x \left (t \right ) &= -{\mathrm e}^{-\frac {\left (-8+\left (244+12 \sqrt {417}\right )^{{2}/{3}}-8 \left (244+12 \sqrt {417}\right )^{{1}/{3}}\right ) t}{12 \left (244+12 \sqrt {417}\right )^{{1}/{3}}}} \sin \left (\frac {\left (\left (244+12 \sqrt {417}\right )^{{2}/{3}}+8\right ) t \sqrt {3}\, 2^{{1}/{3}}}{24 \left (61+3 \sqrt {417}\right )^{{1}/{3}}}\right ) c_{2} +{\mathrm e}^{-\frac {\left (-8+\left (244+12 \sqrt {417}\right )^{{2}/{3}}-8 \left (244+12 \sqrt {417}\right )^{{1}/{3}}\right ) t}{12 \left (244+12 \sqrt {417}\right )^{{1}/{3}}}} \cos \left (\frac {\left (\left (244+12 \sqrt {417}\right )^{{2}/{3}}+8\right ) t \sqrt {3}\, 2^{{1}/{3}}}{24 \left (61+3 \sqrt {417}\right )^{{1}/{3}}}\right ) c_3 +c_{1} {\mathrm e}^{\frac {\left (\left (244+12 \sqrt {417}\right )^{{2}/{3}}+4 \left (244+12 \sqrt {417}\right )^{{1}/{3}}-8\right ) t}{6 \left (244+12 \sqrt {417}\right )^{{1}/{3}}}} \\
y &= \frac {-2 c_{1} \left (\left (244+12 \sqrt {417}\right )^{{2}/{3}}+4 \left (244+12 \sqrt {417}\right )^{{1}/{3}}-8\right ) {\mathrm e}^{\frac {\left (\left (244+12 \sqrt {417}\right )^{{2}/{3}}+4 \left (244+12 \sqrt {417}\right )^{{1}/{3}}-8\right ) t}{6 \left (244+12 \sqrt {417}\right )^{{1}/{3}}}}-c_{2} \left (-8+\left (244+12 \sqrt {417}\right )^{{2}/{3}}-8 \left (244+12 \sqrt {417}\right )^{{1}/{3}}\right ) {\mathrm e}^{-\frac {\left (-8+\left (244+12 \sqrt {417}\right )^{{2}/{3}}-8 \left (244+12 \sqrt {417}\right )^{{1}/{3}}\right ) t}{12 \left (244+12 \sqrt {417}\right )^{{1}/{3}}}} \sin \left (\frac {\sqrt {3}\, \left (\left (244+12 \sqrt {417}\right )^{{2}/{3}}+8\right ) t}{12 \left (244+12 \sqrt {417}\right )^{{1}/{3}}}\right )+c_{2} {\mathrm e}^{-\frac {\left (-8+\left (244+12 \sqrt {417}\right )^{{2}/{3}}-8 \left (244+12 \sqrt {417}\right )^{{1}/{3}}\right ) t}{12 \left (244+12 \sqrt {417}\right )^{{1}/{3}}}} \left (\sqrt {3}\, \left (244+12 \sqrt {417}\right )^{{2}/{3}}+8 \sqrt {3}\right ) \cos \left (\frac {\sqrt {3}\, \left (\left (244+12 \sqrt {417}\right )^{{2}/{3}}+8\right ) t}{12 \left (244+12 \sqrt {417}\right )^{{1}/{3}}}\right )+c_3 \left (-8+\left (244+12 \sqrt {417}\right )^{{2}/{3}}-8 \left (244+12 \sqrt {417}\right )^{{1}/{3}}\right ) {\mathrm e}^{-\frac {\left (-8+\left (244+12 \sqrt {417}\right )^{{2}/{3}}-8 \left (244+12 \sqrt {417}\right )^{{1}/{3}}\right ) t}{12 \left (244+12 \sqrt {417}\right )^{{1}/{3}}}} \cos \left (\frac {\sqrt {3}\, \left (\left (244+12 \sqrt {417}\right )^{{2}/{3}}+8\right ) t}{12 \left (244+12 \sqrt {417}\right )^{{1}/{3}}}\right )+c_3 \,{\mathrm e}^{-\frac {\left (-8+\left (244+12 \sqrt {417}\right )^{{2}/{3}}-8 \left (244+12 \sqrt {417}\right )^{{1}/{3}}\right ) t}{12 \left (244+12 \sqrt {417}\right )^{{1}/{3}}}} \left (\sqrt {3}\, \left (244+12 \sqrt {417}\right )^{{2}/{3}}+8 \sqrt {3}\right ) \sin \left (\frac {\sqrt {3}\, \left (\left (244+12 \sqrt {417}\right )^{{2}/{3}}+8\right ) t}{12 \left (244+12 \sqrt {417}\right )^{{1}/{3}}}\right )+12 c_{1} {\mathrm e}^{\frac {\left (\left (244+12 \sqrt {417}\right )^{{2}/{3}}+4 \left (244+12 \sqrt {417}\right )^{{1}/{3}}-8\right ) t}{6 \left (244+12 \sqrt {417}\right )^{{1}/{3}}}} \left (244+12 \sqrt {417}\right )^{{1}/{3}}-12 c_{2} {\mathrm e}^{-\frac {\left (-8+\left (244+12 \sqrt {417}\right )^{{2}/{3}}-8 \left (244+12 \sqrt {417}\right )^{{1}/{3}}\right ) t}{12 \left (244+12 \sqrt {417}\right )^{{1}/{3}}}} \sin \left (\frac {\sqrt {3}\, \left (\left (244+12 \sqrt {417}\right )^{{2}/{3}}+8\right ) t}{12 \left (244+12 \sqrt {417}\right )^{{1}/{3}}}\right ) \left (244+12 \sqrt {417}\right )^{{1}/{3}}+12 c_3 \,{\mathrm e}^{-\frac {\left (-8+\left (244+12 \sqrt {417}\right )^{{2}/{3}}-8 \left (244+12 \sqrt {417}\right )^{{1}/{3}}\right ) t}{12 \left (244+12 \sqrt {417}\right )^{{1}/{3}}}} \cos \left (\frac {\sqrt {3}\, \left (\left (244+12 \sqrt {417}\right )^{{2}/{3}}+8\right ) t}{12 \left (244+12 \sqrt {417}\right )^{{1}/{3}}}\right ) \left (244+12 \sqrt {417}\right )^{{1}/{3}}}{12 \left (244+12 \sqrt {417}\right )^{{1}/{3}}} \\
\text {Expression too large to display} \\
\end{align*}
✓ Solution by Mathematica
Time used: 0.020 (sec). Leaf size: 503
DSolve[{D[x[t],t]==x[t]-y[t],D[y[t],t]==x[t]+2*z[t],D[z[t],t]==-x[t]+z[t]},{x[t],y[t],z[t]},t,IncludeSingularSolutions -> True]
\begin{align*}
x(t)\to -2 c_3 \text {RootSum}\left [\text {$\#$1}^3-2 \text {$\#$1}^2+2 \text {$\#$1}-3\&,\frac {e^{\text {$\#$1} t}}{3 \text {$\#$1}^2-4 \text {$\#$1}+2}\&\right ]-c_2 \text {RootSum}\left [\text {$\#$1}^3-2 \text {$\#$1}^2+2 \text {$\#$1}-3\&,\frac {\text {$\#$1} e^{\text {$\#$1} t}-e^{\text {$\#$1} t}}{3 \text {$\#$1}^2-4 \text {$\#$1}+2}\&\right ]+c_1 \text {RootSum}\left [\text {$\#$1}^3-2 \text {$\#$1}^2+2 \text {$\#$1}-3\&,\frac {\text {$\#$1}^2 e^{\text {$\#$1} t}-\text {$\#$1} e^{\text {$\#$1} t}}{3 \text {$\#$1}^2-4 \text {$\#$1}+2}\&\right ] \\
y(t)\to c_1 \text {RootSum}\left [\text {$\#$1}^3-2 \text {$\#$1}^2+2 \text {$\#$1}-3\&,\frac {\text {$\#$1} e^{\text {$\#$1} t}-3 e^{\text {$\#$1} t}}{3 \text {$\#$1}^2-4 \text {$\#$1}+2}\&\right ]+2 c_3 \text {RootSum}\left [\text {$\#$1}^3-2 \text {$\#$1}^2+2 \text {$\#$1}-3\&,\frac {\text {$\#$1} e^{\text {$\#$1} t}-e^{\text {$\#$1} t}}{3 \text {$\#$1}^2-4 \text {$\#$1}+2}\&\right ]+c_2 \text {RootSum}\left [\text {$\#$1}^3-2 \text {$\#$1}^2+2 \text {$\#$1}-3\&,\frac {\text {$\#$1}^2 e^{\text {$\#$1} t}-2 \text {$\#$1} e^{\text {$\#$1} t}+e^{\text {$\#$1} t}}{3 \text {$\#$1}^2-4 \text {$\#$1}+2}\&\right ] \\
z(t)\to c_2 \text {RootSum}\left [\text {$\#$1}^3-2 \text {$\#$1}^2+2 \text {$\#$1}-3\&,\frac {e^{\text {$\#$1} t}}{3 \text {$\#$1}^2-4 \text {$\#$1}+2}\&\right ]-c_1 \text {RootSum}\left [\text {$\#$1}^3-2 \text {$\#$1}^2+2 \text {$\#$1}-3\&,\frac {\text {$\#$1} e^{\text {$\#$1} t}}{3 \text {$\#$1}^2-4 \text {$\#$1}+2}\&\right ]+c_3 \text {RootSum}\left [\text {$\#$1}^3-2 \text {$\#$1}^2+2 \text {$\#$1}-3\&,\frac {\text {$\#$1}^2 e^{\text {$\#$1} t}-\text {$\#$1} e^{\text {$\#$1} t}+e^{\text {$\#$1} t}}{3 \text {$\#$1}^2-4 \text {$\#$1}+2}\&\right ] \\
\end{align*}