52.9.3 problem 3
Internal
problem
ID
[8381]
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
:
3
Date
solved
:
Monday, January 27, 2025 at 03:49:54 PM
CAS
classification
:
system_of_ODEs
\begin{align*} x^{\prime }\left (t \right )&=-3 x \left (t \right )+4 y-9 z \left (t \right )\\ y^{\prime }&=6 x \left (t \right )-y\\ z^{\prime }\left (t \right )&=10 x \left (t \right )+4 y+3 z \left (t \right ) \end{align*}
✓ Solution by Maple
Time used: 0.408 (sec). Leaf size: 2254
dsolve([diff(x(t),t)=-3*x(t)+4*y(t)-9*z(t),diff(y(t),t)=6*x(t)-y(t),diff(z(t),t)=10*x(t)+4*y(t)+3*z(t)],singsol=all)
\begin{align*}
\text {Expression too large to display} \\
y &= \sin \left (\frac {\left (\left (4726+306 \sqrt {291}\right )^{{2}/{3}}+170\right ) t \sqrt {3}\, 1156^{{1}/{3}}}{204 \left (139+9 \sqrt {291}\right )^{{1}/{3}}}\right ) {\mathrm e}^{\frac {\left (-170+\left (4726+306 \sqrt {291}\right )^{{2}/{3}}-2 \left (4726+306 \sqrt {291}\right )^{{1}/{3}}\right ) t}{6 \left (4726+306 \sqrt {291}\right )^{{1}/{3}}}} c_{2} +{\mathrm e}^{\frac {\left (-170+\left (4726+306 \sqrt {291}\right )^{{2}/{3}}-2 \left (4726+306 \sqrt {291}\right )^{{1}/{3}}\right ) t}{6 \left (4726+306 \sqrt {291}\right )^{{1}/{3}}}} \cos \left (\frac {\left (\left (4726+306 \sqrt {291}\right )^{{2}/{3}}+170\right ) t \sqrt {3}\, 1156^{{1}/{3}}}{204 \left (139+9 \sqrt {291}\right )^{{1}/{3}}}\right ) c_3 +c_{1} {\mathrm e}^{-\frac {\left (\left (4726+306 \sqrt {291}\right )^{{2}/{3}}+\left (4726+306 \sqrt {291}\right )^{{1}/{3}}-170\right ) t}{3 \left (4726+306 \sqrt {291}\right )^{{1}/{3}}}} \\
\text {Expression too large to display} \\
\end{align*}
✓ Solution by Mathematica
Time used: 0.022 (sec). Leaf size: 510
DSolve[{D[x[t],t]==-3*x[t]+4*y[t]-9*z[t],D[y[t],t]==6*x[t]-y[t],D[z[t],t]==10*x[t]+4*y[t]+3*z[t]},{x[t],y[t],z[t]},t,IncludeSingularSolutions -> True]
\begin{align*}
x(t)\to 4 c_2 \text {RootSum}\left [\text {$\#$1}^3+\text {$\#$1}^2+57 \text {$\#$1}+369\&,\frac {\text {$\#$1} e^{\text {$\#$1} t}-12 e^{\text {$\#$1} t}}{3 \text {$\#$1}^2+2 \text {$\#$1}+57}\&\right ]-9 c_3 \text {RootSum}\left [\text {$\#$1}^3+\text {$\#$1}^2+57 \text {$\#$1}+369\&,\frac {\text {$\#$1} e^{\text {$\#$1} t}+e^{\text {$\#$1} t}}{3 \text {$\#$1}^2+2 \text {$\#$1}+57}\&\right ]+c_1 \text {RootSum}\left [\text {$\#$1}^3+\text {$\#$1}^2+57 \text {$\#$1}+369\&,\frac {\text {$\#$1}^2 e^{\text {$\#$1} t}-2 \text {$\#$1} e^{\text {$\#$1} t}-3 e^{\text {$\#$1} t}}{3 \text {$\#$1}^2+2 \text {$\#$1}+57}\&\right ] \\
y(t)\to -54 c_3 \text {RootSum}\left [\text {$\#$1}^3+\text {$\#$1}^2+57 \text {$\#$1}+369\&,\frac {e^{\text {$\#$1} t}}{3 \text {$\#$1}^2+2 \text {$\#$1}+57}\&\right ]+6 c_1 \text {RootSum}\left [\text {$\#$1}^3+\text {$\#$1}^2+57 \text {$\#$1}+369\&,\frac {\text {$\#$1} e^{\text {$\#$1} t}-3 e^{\text {$\#$1} t}}{3 \text {$\#$1}^2+2 \text {$\#$1}+57}\&\right ]+c_2 \text {RootSum}\left [\text {$\#$1}^3+\text {$\#$1}^2+57 \text {$\#$1}+369\&,\frac {\text {$\#$1}^2 e^{\text {$\#$1} t}+81 e^{\text {$\#$1} t}}{3 \text {$\#$1}^2+2 \text {$\#$1}+57}\&\right ] \\
z(t)\to 4 c_2 \text {RootSum}\left [\text {$\#$1}^3+\text {$\#$1}^2+57 \text {$\#$1}+369\&,\frac {\text {$\#$1} e^{\text {$\#$1} t}+13 e^{\text {$\#$1} t}}{3 \text {$\#$1}^2+2 \text {$\#$1}+57}\&\right ]+2 c_1 \text {RootSum}\left [\text {$\#$1}^3+\text {$\#$1}^2+57 \text {$\#$1}+369\&,\frac {5 \text {$\#$1} e^{\text {$\#$1} t}+17 e^{\text {$\#$1} t}}{3 \text {$\#$1}^2+2 \text {$\#$1}+57}\&\right ]+c_3 \text {RootSum}\left [\text {$\#$1}^3+\text {$\#$1}^2+57 \text {$\#$1}+369\&,\frac {\text {$\#$1}^2 e^{\text {$\#$1} t}+4 \text {$\#$1} e^{\text {$\#$1} t}-21 e^{\text {$\#$1} t}}{3 \text {$\#$1}^2+2 \text {$\#$1}+57}\&\right ] \\
\end{align*}