14.29.15 problem 16
Internal
problem
ID
[2813]
Book
:
Differential
equations
and
their
applications,
4th
ed.,
M.
Braun
Section
:
Chapter
4.
Qualitative
theory
of
differential
equations.
Section
4.2
(Stability
of
linear
systems).
Page
383
Problem
number
:
16
Date
solved
:
Tuesday, March 04, 2025 at 02:43:05 PM
CAS
classification
:
system_of_ODEs
\begin{align*} \frac {d}{d t}x_{1} \left (t \right )&=-x_{1} \left (t \right )-x_{2} \left (t \right )+1\\ \frac {d}{d t}x_{2} \left (t \right )&=2 x_{1} \left (t \right )-x_{2} \left (t \right )+5 \end{align*}
✓ Maple. Time used: 0.038 (sec). Leaf size: 60
ode:=[diff(x__1(t),t) = -x__1(t)-x__2(t)+1, diff(x__2(t),t) = 2*x__1(t)-x__2(t)+5];
dsolve(ode);
\begin{align*}
x_{1} \left (t \right ) &= -\frac {4}{3}+{\mathrm e}^{-t} \left (\cos \left (\sqrt {2}\, t \right ) c_1 +c_2 \sin \left (\sqrt {2}\, t \right )\right ) \\
x_{2} \left (t \right ) &= \frac {7}{3}-{\mathrm e}^{-t} \sqrt {2}\, \left (c_2 \cos \left (\sqrt {2}\, t \right )-c_1 \sin \left (\sqrt {2}\, t \right )\right ) \\
\end{align*}
✓ Mathematica. Time used: 0.07 (sec). Leaf size: 91
ode={D[x1[t],t]==-x1[t]-x2[t],D[x2[t],t]==2*x1[t]-x2[t]+5};
ic={};
DSolve[{ode,ic},{x1[t],x2[t]},t,IncludeSingularSolutions->True]
\begin{align*}
\text {x1}(t)\to c_1 e^{-t} \cos \left (\sqrt {2} t\right )-\frac {c_2 e^{-t} \sin \left (\sqrt {2} t\right )}{\sqrt {2}}-\frac {5}{3} \\
\text {x2}(t)\to c_2 e^{-t} \cos \left (\sqrt {2} t\right )+\sqrt {2} c_1 e^{-t} \sin \left (\sqrt {2} t\right )+\frac {5}{3} \\
\end{align*}
✓ Sympy. Time used: 0.370 (sec). Leaf size: 128
from sympy import *
t = symbols("t")
x__1 = Function("x__1")
x__2 = Function("x__2")
ode=[Eq(x__1(t) + x__2(t) + Derivative(x__1(t), t) - 1,0),Eq(-2*x__1(t) + x__2(t) + Derivative(x__2(t), t) - 5,0)]
ics = {}
dsolve(ode,func=[x__1(t),x__2(t)],ics=ics)
\[
\left [ x^{1}{\left (t \right )} = - \frac {\sqrt {2} C_{1} e^{- t} \sin {\left (\sqrt {2} t \right )}}{2} - \frac {\sqrt {2} C_{2} e^{- t} \cos {\left (\sqrt {2} t \right )}}{2} - \frac {4 \sin ^{2}{\left (\sqrt {2} t \right )}}{3} - \frac {4 \cos ^{2}{\left (\sqrt {2} t \right )}}{3}, \ x^{2}{\left (t \right )} = C_{1} e^{- t} \cos {\left (\sqrt {2} t \right )} - C_{2} e^{- t} \sin {\left (\sqrt {2} t \right )} + \frac {7 \sin ^{2}{\left (\sqrt {2} t \right )}}{3} + \frac {7 \cos ^{2}{\left (\sqrt {2} t \right )}}{3}\right ]
\]