72.9.6 problem 6
Internal
problem
ID
[14723]
Book
:
DIFFERENTIAL
EQUATIONS
by
Paul
Blanchard,
Robert
L.
Devaney,
Glen
R.
Hall.
4th
edition.
Brooks/Cole.
Boston,
USA.
2012
Section
:
Chapter
3.
Linear
Systems.
Exercises
section
3.1.
page
258
Problem
number
:
6
Date
solved
:
Thursday, March 13, 2025 at 04:16:29 AM
CAS
classification
:
system_of_ODEs
\begin{align*} \frac {d}{d t}x \left (t \right )&=3 y \left (t \right )\\ \frac {d}{d t}y \left (t \right )&=3 \pi y \left (t \right )-\frac {x \left (t \right )}{3} \end{align*}
✓ Maple. Time used: 0.053 (sec). Leaf size: 119
ode:=[diff(x(t),t) = 3*y(t), diff(y(t),t) = 3*Pi*y(t)-1/3*x(t)];
dsolve(ode);
\begin{align*}
x \left (t \right ) &= c_{1} {\mathrm e}^{\frac {\left (3 \pi -\sqrt {9 \pi ^{2}-4}\right ) t}{2}}+c_{2} {\mathrm e}^{\frac {\left (3 \pi +\sqrt {9 \pi ^{2}-4}\right ) t}{2}} \\
y &= \left (\frac {\pi }{2}+\frac {\sqrt {9 \pi ^{2}-4}}{6}\right ) c_{2} {\mathrm e}^{\frac {\left (3 \pi +\sqrt {9 \pi ^{2}-4}\right ) t}{2}}+\left (\frac {\pi }{2}-\frac {\sqrt {9 \pi ^{2}-4}}{6}\right ) c_{1} {\mathrm e}^{\frac {\left (3 \pi -\sqrt {9 \pi ^{2}-4}\right ) t}{2}} \\
\end{align*}
✓ Mathematica. Time used: 0.009 (sec). Leaf size: 233
ode={D[x[t],t]==3*y[t],D[y[t],t]==3*Pi*y[t]-1/3*x[t]};
ic={};
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
\begin{align*}
x(t)\to \frac {e^{-\frac {1}{2} \left (\sqrt {9 \pi ^2-4}-3 \pi \right ) t} \left (\sqrt {9 \pi ^2-4} c_1 \left (e^{\sqrt {9 \pi ^2-4} t}+1\right )-3 \pi c_1 \left (e^{\sqrt {9 \pi ^2-4} t}-1\right )+6 c_2 \left (e^{\sqrt {9 \pi ^2-4} t}-1\right )\right )}{2 \sqrt {9 \pi ^2-4}} \\
y(t)\to \frac {e^{-\frac {1}{2} \left (\sqrt {9 \pi ^2-4}-3 \pi \right ) t} \left (3 c_2 \left (3 \pi \left (e^{\sqrt {9 \pi ^2-4} t}-1\right )+\sqrt {9 \pi ^2-4} \left (e^{\sqrt {9 \pi ^2-4} t}+1\right )\right )-2 c_1 \left (e^{\sqrt {9 \pi ^2-4} t}-1\right )\right )}{6 \sqrt {9 \pi ^2-4}} \\
\end{align*}
✓ Sympy. Time used: 0.387 (sec). Leaf size: 304
from sympy import *
t = symbols("t")
x = Function("x")
y = Function("y")
ode=[Eq(-3*y(t) + Derivative(x(t), t),0),Eq(x(t)/3 - 3*pi*y(t) + Derivative(y(t), t),0)]
ics = {}
dsolve(ode,func=[x(t),y(t)],ics=ics)
\[
\left [ x{\left (t \right )} = - \frac {3 C_{1} \left (2 \sqrt {-4 + 9 \pi ^{2}} + 6 \pi + \sqrt {-4 + 9 \pi ^{2}} \sqrt {- 72 \pi ^{2} + 16 + 81 \pi ^{4}} + 3 \pi \sqrt {- 72 \pi ^{2} + 16 + 81 \pi ^{4}} - \sqrt {-4 + 9 \pi ^{2}} \left (2 - 9 \pi ^{2}\right ) - 3 \pi \left (2 - 9 \pi ^{2}\right )\right ) e^{\frac {t \left (- \sqrt {-2 + 3 \pi } \sqrt {2 + 3 \pi } + 3 \pi \right )}{2}}}{4 \left (2 - 9 \pi ^{2}\right )} - \frac {3 C_{2} \left (\sqrt {-4 + 9 \pi ^{2}} \left (2 - 9 \pi ^{2}\right ) - \sqrt {-4 + 9 \pi ^{2}} \sqrt {- 72 \pi ^{2} + 16 + 81 \pi ^{4}} - 2 \sqrt {-4 + 9 \pi ^{2}} + 6 \pi + 3 \pi \sqrt {- 72 \pi ^{2} + 16 + 81 \pi ^{4}} - 3 \pi \left (2 - 9 \pi ^{2}\right )\right ) e^{\frac {t \left (\sqrt {-2 + 3 \pi } \sqrt {2 + 3 \pi } + 3 \pi \right )}{2}}}{4 \left (2 - 9 \pi ^{2}\right )}, \ y{\left (t \right )} = C_{1} e^{\frac {t \left (- \sqrt {-2 + 3 \pi } \sqrt {2 + 3 \pi } + 3 \pi \right )}{2}} + C_{2} e^{\frac {t \left (\sqrt {-2 + 3 \pi } \sqrt {2 + 3 \pi } + 3 \pi \right )}{2}}\right ]
\]