50.29.10 problem 3(f)
Internal
problem
ID
[8206]
Book
:
Differential
Equations:
Theory,
Technique,
and
Practice
by
George
Simmons,
Steven
Krantz.
McGraw-Hill
NY.
2007.
1st
Edition.
Section
:
Chapter
10.
Systems
of
First-Order
Equations.
Section
A.
Drill
exercises.
Page
400
Problem
number
:
3(f)
Date
solved
:
Monday, January 27, 2025 at 03:45:49 PM
CAS
classification
:
system_of_ODEs
\begin{align*} x^{\prime }\left (t \right )&=-x \left (t \right )+y \left (t \right )-z \left (t \right )\\ y^{\prime }\left (t \right )&=2 x \left (t \right )-y \left (t \right )-4 z \left (t \right )\\ z^{\prime }\left (t \right )&=3 x \left (t \right )-y \left (t \right )+z \left (t \right ) \end{align*}
✓ Solution by Maple
Time used: 0.233 (sec). Leaf size: 3192
dsolve([diff(x(t),t)=-x(t)+y(t)-z(t),diff(y(t),t)=2*x(t)-y(t)-4*z(t),diff(z(t),t)=3*x(t)-y(t)+z(t)],singsol=all)
\begin{align*}
x \left (t \right ) &= c_{2} {\mathrm e}^{\frac {\left (13+\left (154+3 \sqrt {2391}\right )^{{2}/{3}}-2 \left (154+3 \sqrt {2391}\right )^{{1}/{3}}\right ) t}{6 \left (154+3 \sqrt {2391}\right )^{{1}/{3}}}} \sin \left (\frac {\sqrt {3}\, \left (\left (154+3 \sqrt {2391}\right )^{{2}/{3}}-13\right ) t}{6 \left (154+3 \sqrt {2391}\right )^{{1}/{3}}}\right )+c_3 \,{\mathrm e}^{\frac {\left (13+\left (154+3 \sqrt {2391}\right )^{{2}/{3}}-2 \left (154+3 \sqrt {2391}\right )^{{1}/{3}}\right ) t}{6 \left (154+3 \sqrt {2391}\right )^{{1}/{3}}}} \cos \left (\frac {\sqrt {3}\, \left (\left (154+3 \sqrt {2391}\right )^{{2}/{3}}-13\right ) t}{6 \left (154+3 \sqrt {2391}\right )^{{1}/{3}}}\right )+c_{1} {\mathrm e}^{-\frac {\left (\left (154+3 \sqrt {2391}\right )^{{2}/{3}}+\left (154+3 \sqrt {2391}\right )^{{1}/{3}}+13\right ) t}{3 \left (154+3 \sqrt {2391}\right )^{{1}/{3}}}} \\
\text {Expression too large to display} \\
\text {Expression too large to display} \\
\end{align*}
✓ Solution by Mathematica
Time used: 0.019 (sec). Leaf size: 501
DSolve[{D[x[t],t]==-x[t]+y[t]-z[t],D[y[t],t]==2*x[t]-y[t]-4*z[t],D[z[t],t]==3*x[t]-y[t]+z[t]},{x[t],y[t],z[t]},t,IncludeSingularSolutions -> True]
\begin{align*}
x(t)\to c_2 \text {RootSum}\left [\text {$\#$1}^3+\text {$\#$1}^2-4 \text {$\#$1}+10\&,\frac {\text {$\#$1} e^{\text {$\#$1} t}}{3 \text {$\#$1}^2+2 \text {$\#$1}-4}\&\right ]-c_3 \text {RootSum}\left [\text {$\#$1}^3+\text {$\#$1}^2-4 \text {$\#$1}+10\&,\frac {\text {$\#$1} e^{\text {$\#$1} t}+5 e^{\text {$\#$1} t}}{3 \text {$\#$1}^2+2 \text {$\#$1}-4}\&\right ]+c_1 \text {RootSum}\left [\text {$\#$1}^3+\text {$\#$1}^2-4 \text {$\#$1}+10\&,\frac {\text {$\#$1}^2 e^{\text {$\#$1} t}-5 e^{\text {$\#$1} t}}{3 \text {$\#$1}^2+2 \text {$\#$1}-4}\&\right ] \\
y(t)\to 2 c_1 \text {RootSum}\left [\text {$\#$1}^3+\text {$\#$1}^2-4 \text {$\#$1}+10\&,\frac {\text {$\#$1} e^{\text {$\#$1} t}-7 e^{\text {$\#$1} t}}{3 \text {$\#$1}^2+2 \text {$\#$1}-4}\&\right ]-2 c_3 \text {RootSum}\left [\text {$\#$1}^3+\text {$\#$1}^2-4 \text {$\#$1}+10\&,\frac {2 \text {$\#$1} e^{\text {$\#$1} t}+3 e^{\text {$\#$1} t}}{3 \text {$\#$1}^2+2 \text {$\#$1}-4}\&\right ]+c_2 \text {RootSum}\left [\text {$\#$1}^3+\text {$\#$1}^2-4 \text {$\#$1}+10\&,\frac {\text {$\#$1}^2 e^{\text {$\#$1} t}+2 e^{\text {$\#$1} t}}{3 \text {$\#$1}^2+2 \text {$\#$1}-4}\&\right ] \\
z(t)\to -c_2 \text {RootSum}\left [\text {$\#$1}^3+\text {$\#$1}^2-4 \text {$\#$1}+10\&,\frac {\text {$\#$1} e^{\text {$\#$1} t}-2 e^{\text {$\#$1} t}}{3 \text {$\#$1}^2+2 \text {$\#$1}-4}\&\right ]+c_1 \text {RootSum}\left [\text {$\#$1}^3+\text {$\#$1}^2-4 \text {$\#$1}+10\&,\frac {3 \text {$\#$1} e^{\text {$\#$1} t}+e^{\text {$\#$1} t}}{3 \text {$\#$1}^2+2 \text {$\#$1}-4}\&\right ]+c_3 \text {RootSum}\left [\text {$\#$1}^3+\text {$\#$1}^2-4 \text {$\#$1}+10\&,\frac {\text {$\#$1}^2 e^{\text {$\#$1} t}+2 \text {$\#$1} e^{\text {$\#$1} t}-e^{\text {$\#$1} t}}{3 \text {$\#$1}^2+2 \text {$\#$1}-4}\&\right ] \\
\end{align*}