14.32.7 problem 7
Internal
problem
ID
[2831]
Book
:
Differential
equations
and
their
applications,
4th
ed.,
M.
Braun
Section
:
Chapter
4.
Qualitative
theory
of
differential
equations.
Section
4.7
(Phase
portraits
of
linear
systems).
Page
427
Problem
number
:
7
Date
solved
:
Sunday, March 30, 2025 at 12:33:27 AM
CAS
classification
:
system_of_ODEs
\begin{align*} \frac {d}{d t}x_{1} \left (t \right )&=2 x_{2} \left (t \right )\\ \frac {d}{d t}x_{2} \left (t \right )&=-2 x_{1} \left (t \right )-x_{2} \left (t \right ) \end{align*}
✓ Maple. Time used: 0.136 (sec). Leaf size: 81
ode:=[diff(x__1(t),t) = 2*x__2(t), diff(x__2(t),t) = -2*x__1(t)-x__2(t)];
dsolve(ode);
\begin{align*}
x_{1} \left (t \right ) &= {\mathrm e}^{-\frac {t}{2}} \left (\sin \left (\frac {\sqrt {15}\, t}{2}\right ) c_1 +\cos \left (\frac {\sqrt {15}\, t}{2}\right ) c_2 \right ) \\
x_{2} \left (t \right ) &= -\frac {{\mathrm e}^{-\frac {t}{2}} \left (\sin \left (\frac {\sqrt {15}\, t}{2}\right ) \sqrt {15}\, c_2 -\cos \left (\frac {\sqrt {15}\, t}{2}\right ) \sqrt {15}\, c_1 +\sin \left (\frac {\sqrt {15}\, t}{2}\right ) c_1 +\cos \left (\frac {\sqrt {15}\, t}{2}\right ) c_2 \right )}{4} \\
\end{align*}
✓ Mathematica. Time used: 0.023 (sec). Leaf size: 111
ode={D[x1[t],t]==0*x1[t]+2*x2[t],D[x2[t],t]==-2*x1[t]-1*x2[t]};
ic={};
DSolve[{ode,ic},{x1[t],x2[t]},t,IncludeSingularSolutions->True]
\begin{align*}
\text {x1}(t)\to \frac {1}{15} e^{-t/2} \left (15 c_1 \cos \left (\frac {\sqrt {15} t}{2}\right )+\sqrt {15} (c_1+4 c_2) \sin \left (\frac {\sqrt {15} t}{2}\right )\right ) \\
\text {x2}(t)\to \frac {1}{15} e^{-t/2} \left (15 c_2 \cos \left (\frac {\sqrt {15} t}{2}\right )-\sqrt {15} (4 c_1+c_2) \sin \left (\frac {\sqrt {15} t}{2}\right )\right ) \\
\end{align*}
✓ Sympy. Time used: 0.190 (sec). Leaf size: 92
from sympy import *
t = symbols("t")
x__1 = Function("x__1")
x__2 = Function("x__2")
ode=[Eq(-2*x__2(t) + Derivative(x__1(t), t),0),Eq(2*x__1(t) + x__2(t) + Derivative(x__2(t), t),0)]
ics = {}
dsolve(ode,func=[x__1(t),x__2(t)],ics=ics)
\[
\left [ x^{1}{\left (t \right )} = - \left (\frac {C_{1}}{4} - \frac {\sqrt {15} C_{2}}{4}\right ) e^{- \frac {t}{2}} \cos {\left (\frac {\sqrt {15} t}{2} \right )} + \left (\frac {\sqrt {15} C_{1}}{4} + \frac {C_{2}}{4}\right ) e^{- \frac {t}{2}} \sin {\left (\frac {\sqrt {15} t}{2} \right )}, \ x^{2}{\left (t \right )} = C_{1} e^{- \frac {t}{2}} \cos {\left (\frac {\sqrt {15} t}{2} \right )} - C_{2} e^{- \frac {t}{2}} \sin {\left (\frac {\sqrt {15} t}{2} \right )}\right ]
\]