76.27.11 problem 11
Internal
problem
ID
[17786]
Book
:
Differential
equations.
An
introduction
to
modern
methods
and
applications.
James
Brannan,
William
E.
Boyce.
Third
edition.
Wiley
2015
Section
:
Chapter
6.
Systems
of
First
Order
Linear
Equations.
Section
6.5
(Fundamental
Matrices
and
the
Exponential
of
a
Matrix).
Problems
at
page
430
Problem
number
:
11
Date
solved
:
Monday, March 31, 2025 at 04:27:40 PM
CAS
classification
:
system_of_ODEs
\begin{align*} \frac {d}{d t}x_{1} \left (t \right )&=-3 x_{1} \left (t \right )+4 x_{2} \left (t \right )\\ \frac {d}{d t}x_{2} \left (t \right )&=-x_{1} \left (t \right )-2 x_{2} \left (t \right ) \end{align*}
✓ Maple. Time used: 0.147 (sec). Leaf size: 83
ode:=[diff(x__1(t),t) = -3*x__1(t)+4*x__2(t), diff(x__2(t),t) = -x__1(t)-2*x__2(t)];
dsolve(ode);
\begin{align*}
x_{1} \left (t \right ) &= {\mathrm e}^{-\frac {5 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 {5 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 )}{8} \\
\end{align*}
✓ Mathematica. Time used: 0.019 (sec). Leaf size: 111
ode={D[x1[t],t]==-3*x1[t]+4*x2[t],D[x2[t],t]==-1*x1[t]-2*x2[t]};
ic={};
DSolve[{ode,ic},{x1[t],x2[t]},t,IncludeSingularSolutions->True]
\begin{align*}
\text {x1}(t)\to \frac {1}{15} e^{-5 t/2} \left (15 c_1 \cos \left (\frac {\sqrt {15} t}{2}\right )-\sqrt {15} (c_1-8 c_2) \sin \left (\frac {\sqrt {15} t}{2}\right )\right ) \\
\text {x2}(t)\to \frac {1}{15} e^{-5 t/2} \left (15 c_2 \cos \left (\frac {\sqrt {15} t}{2}\right )+\sqrt {15} (c_2-2 c_1) \sin \left (\frac {\sqrt {15} t}{2}\right )\right ) \\
\end{align*}
✓ Sympy. Time used: 0.189 (sec). Leaf size: 99
from sympy import *
t = symbols("t")
x__1 = Function("x__1")
x__2 = Function("x__2")
ode=[Eq(3*x__1(t) - 4*x__2(t) + Derivative(x__1(t), t),0),Eq(x__1(t) + 2*x__2(t) + Derivative(x__2(t), t),0)]
ics = {}
dsolve(ode,func=[x__1(t),x__2(t)],ics=ics)
\[
\left [ x^{1}{\left (t \right )} = \left (\frac {C_{1}}{2} + \frac {\sqrt {15} C_{2}}{2}\right ) e^{- \frac {5 t}{2}} \cos {\left (\frac {\sqrt {15} t}{2} \right )} + \left (\frac {\sqrt {15} C_{1}}{2} - \frac {C_{2}}{2}\right ) e^{- \frac {5 t}{2}} \sin {\left (\frac {\sqrt {15} t}{2} \right )}, \ x^{2}{\left (t \right )} = C_{1} e^{- \frac {5 t}{2}} \cos {\left (\frac {\sqrt {15} t}{2} \right )} - C_{2} e^{- \frac {5 t}{2}} \sin {\left (\frac {\sqrt {15} t}{2} \right )}\right ]
\]