20.14.2 problem 13
Internal
problem
ID
[3826]
Book
:
Differential
equations
and
linear
algebra,
Stephen
W.
Goode
and
Scott
A
Annin.
Fourth
edition,
2015
Section
:
Chapter
9,
First
order
linear
systems.
Section
9.2,
page
592
Problem
number
:
13
Date
solved
:
Tuesday, March 04, 2025 at 05:17:38 PM
CAS
classification
:
system_of_ODEs
\begin{align*} \frac {d}{d t}x_{1} \left (t \right )&=x_{2} \left (t \right )\\ \frac {d}{d t}x_{2} \left (t \right )&=-b x_{1} \left (t \right )-a x_{2} \left (t \right ) \end{align*}
✓ Maple. Time used: 0.046 (sec). Leaf size: 115
ode:=[diff(x__1(t),t) = x__2(t), diff(x__2(t),t) = -b*x__1(t)-a*x__2(t)];
dsolve(ode);
\begin{align*}
x_{1} \left (t \right ) &= c_{1} {\mathrm e}^{\frac {\left (-a +\sqrt {a^{2}-4 b}\right ) t}{2}}+c_{2} {\mathrm e}^{-\frac {\left (a +\sqrt {a^{2}-4 b}\right ) t}{2}} \\
x_{2} \left (t \right ) &= \frac {c_{1} \left (-a +\sqrt {a^{2}-4 b}\right ) {\mathrm e}^{\frac {\left (-a +\sqrt {a^{2}-4 b}\right ) t}{2}}}{2}+\frac {c_{2} \left (-a -\sqrt {a^{2}-4 b}\right ) {\mathrm e}^{\frac {\left (-a -\sqrt {a^{2}-4 b}\right ) t}{2}}}{2} \\
\end{align*}
✓ Mathematica. Time used: 0.012 (sec). Leaf size: 227
ode={D[x1[t],t]==x2[t],D[x2[t],t]==-b*x1[t]-a*x2[t]};
ic={};
DSolve[{ode,ic},{x1[t],x2[t]},t,IncludeSingularSolutions->True]
\begin{align*}
\text {x1}(t)\to \frac {e^{-\frac {1}{2} t \left (\sqrt {a^2-4 b}+a\right )} \left (a c_1 \left (e^{t \sqrt {a^2-4 b}}-1\right )+c_1 \sqrt {a^2-4 b} \left (e^{t \sqrt {a^2-4 b}}+1\right )+2 c_2 \left (e^{t \sqrt {a^2-4 b}}-1\right )\right )}{2 \sqrt {a^2-4 b}} \\
\text {x2}(t)\to \frac {e^{-\frac {1}{2} t \left (\sqrt {a^2-4 b}+a\right )} \left (c_2 \left (a \left (-e^{t \sqrt {a^2-4 b}}\right )+\sqrt {a^2-4 b} \left (e^{t \sqrt {a^2-4 b}}+1\right )+a\right )-2 b c_1 \left (e^{t \sqrt {a^2-4 b}}-1\right )\right )}{2 \sqrt {a^2-4 b}} \\
\end{align*}
✓ Sympy. Time used: 0.274 (sec). Leaf size: 107
from sympy import *
t = symbols("t")
a = symbols("a")
b = symbols("b")
x__1 = Function("x__1")
x__2 = Function("x__2")
ode=[Eq(-x__2(t) + Derivative(x__1(t), t),0),Eq(a*x__2(t) + b*x__1(t) + Derivative(x__2(t), t),0)]
ics = {}
dsolve(ode,func=[x__1(t),x__2(t)],ics=ics)
\[
\left [ x^{1}{\left (t \right )} = - \frac {2 C_{1} e^{- \frac {t \left (a - \sqrt {a^{2} - 4 b}\right )}{2}}}{a - \sqrt {a^{2} - 4 b}} - \frac {2 C_{2} e^{- \frac {t \left (a + \sqrt {a^{2} - 4 b}\right )}{2}}}{a + \sqrt {a^{2} - 4 b}}, \ x^{2}{\left (t \right )} = C_{1} e^{- \frac {t \left (a - \sqrt {a^{2} - 4 b}\right )}{2}} + C_{2} e^{- \frac {t \left (a + \sqrt {a^{2} - 4 b}\right )}{2}}\right ]
\]