14.32.5 problem 5
Internal
problem
ID
[2829]
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
:
5
Date
solved
:
Sunday, March 30, 2025 at 12:33:24 AM
CAS
classification
:
system_of_ODEs
\begin{align*} \frac {d}{d t}x_{1} \left (t \right )&=x_{1} \left (t \right )-4 x_{2} \left (t \right )\\ \frac {d}{d t}x_{2} \left (t \right )&=-8 x_{1} \left (t \right )+4 x_{2} \left (t \right ) \end{align*}
✓ Maple. Time used: 0.119 (sec). Leaf size: 85
ode:=[diff(x__1(t),t) = x__1(t)-4*x__2(t), diff(x__2(t),t) = -8*x__1(t)+4*x__2(t)];
dsolve(ode);
\begin{align*}
x_{1} \left (t \right ) &= c_1 \,{\mathrm e}^{\frac {\left (5+\sqrt {137}\right ) t}{2}}+c_2 \,{\mathrm e}^{-\frac {\left (-5+\sqrt {137}\right ) t}{2}} \\
x_{2} \left (t \right ) &= -\frac {c_1 \,{\mathrm e}^{\frac {\left (5+\sqrt {137}\right ) t}{2}} \sqrt {137}}{8}+\frac {c_2 \,{\mathrm e}^{-\frac {\left (-5+\sqrt {137}\right ) t}{2}} \sqrt {137}}{8}-\frac {3 c_1 \,{\mathrm e}^{\frac {\left (5+\sqrt {137}\right ) t}{2}}}{8}-\frac {3 c_2 \,{\mathrm e}^{-\frac {\left (-5+\sqrt {137}\right ) t}{2}}}{8} \\
\end{align*}
✓ Mathematica. Time used: 0.009 (sec). Leaf size: 148
ode={D[x1[t],t]==1*x1[t]-4*x2[t],D[x2[t],t]==-8*x1[t]+4*x2[t]};
ic={};
DSolve[{ode,ic},{x1[t],x2[t]},t,IncludeSingularSolutions->True]
\begin{align*}
\text {x1}(t)\to \frac {1}{274} e^{-\frac {1}{2} \left (\sqrt {137}-5\right ) t} \left (c_1 \left (\left (137-3 \sqrt {137}\right ) e^{\sqrt {137} t}+137+3 \sqrt {137}\right )-8 \sqrt {137} c_2 \left (e^{\sqrt {137} t}-1\right )\right ) \\
\text {x2}(t)\to \frac {1}{274} e^{-\frac {1}{2} \left (\sqrt {137}-5\right ) t} \left (c_2 \left (\left (137+3 \sqrt {137}\right ) e^{\sqrt {137} t}+137-3 \sqrt {137}\right )-16 \sqrt {137} c_1 \left (e^{\sqrt {137} t}-1\right )\right ) \\
\end{align*}
✓ Sympy. Time used: 0.205 (sec). Leaf size: 75
from sympy import *
t = symbols("t")
x__1 = Function("x__1")
x__2 = Function("x__2")
ode=[Eq(-x__1(t) + 4*x__2(t) + Derivative(x__1(t), t),0),Eq(8*x__1(t) - 4*x__2(t) + Derivative(x__2(t), t),0)]
ics = {}
dsolve(ode,func=[x__1(t),x__2(t)],ics=ics)
\[
\left [ x^{1}{\left (t \right )} = \frac {C_{1} \left (3 - \sqrt {137}\right ) e^{\frac {t \left (5 + \sqrt {137}\right )}{2}}}{16} + \frac {C_{2} \left (3 + \sqrt {137}\right ) e^{\frac {t \left (5 - \sqrt {137}\right )}{2}}}{16}, \ x^{2}{\left (t \right )} = C_{1} e^{\frac {t \left (5 + \sqrt {137}\right )}{2}} + C_{2} e^{\frac {t \left (5 - \sqrt {137}\right )}{2}}\right ]
\]