76.6.5 problem 5
Internal
problem
ID
[17389]
Book
:
Differential
equations.
An
introduction
to
modern
methods
and
applications.
James
Brannan,
William
E.
Boyce.
Third
edition.
Wiley
2015
Section
:
Chapter
3.
Systems
of
two
first
order
equations.
Section
3.2
(Two
first
order
linear
differential
equations).
Problems
at
page
142
Problem
number
:
5
Date
solved
:
Monday, March 31, 2025 at 04:12:38 PM
CAS
classification
:
system_of_ODEs
\begin{align*} \frac {d}{d t}x \left (t \right )&=3 x \left (t \right )-y \left (t \right )\\ \frac {d}{d t}y \left (t \right )&=x \left (t \right )+2 y \left (t \right ) \end{align*}
✓ Maple. Time used: 0.137 (sec). Leaf size: 81
ode:=[diff(x(t),t) = 3*x(t)-y(t), diff(y(t),t) = x(t)+2*y(t)];
dsolve(ode);
\begin{align*}
x \left (t \right ) &= {\mathrm e}^{\frac {5 t}{2}} \left (\sin \left (\frac {\sqrt {3}\, t}{2}\right ) c_1 +\cos \left (\frac {\sqrt {3}\, t}{2}\right ) c_2 \right ) \\
y \left (t \right ) &= \frac {{\mathrm e}^{\frac {5 t}{2}} \left (\sin \left (\frac {\sqrt {3}\, t}{2}\right ) \sqrt {3}\, c_2 -\cos \left (\frac {\sqrt {3}\, t}{2}\right ) \sqrt {3}\, c_1 +\sin \left (\frac {\sqrt {3}\, t}{2}\right ) c_1 +\cos \left (\frac {\sqrt {3}\, t}{2}\right ) c_2 \right )}{2} \\
\end{align*}
✓ Mathematica. Time used: 0.017 (sec). Leaf size: 112
ode={D[x[t],t]==3*x[t]-y[t],D[y[t],t]==x[t]+2*y[t]};
ic={};
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
\begin{align*}
x(t)\to \frac {1}{3} e^{5 t/2} \left (3 c_1 \cos \left (\frac {\sqrt {3} t}{2}\right )+\sqrt {3} (c_1-2 c_2) \sin \left (\frac {\sqrt {3} t}{2}\right )\right ) \\
y(t)\to \frac {1}{3} e^{5 t/2} \left (3 c_2 \cos \left (\frac {\sqrt {3} t}{2}\right )+\sqrt {3} (2 c_1-c_2) \sin \left (\frac {\sqrt {3} t}{2}\right )\right ) \\
\end{align*}
✓ Sympy. Time used: 0.189 (sec). Leaf size: 99
from sympy import *
t = symbols("t")
x = Function("x")
y = Function("y")
ode=[Eq(-3*x(t) + y(t) + Derivative(x(t), t),0),Eq(-x(t) - 2*y(t) + Derivative(y(t), t),0)]
ics = {}
dsolve(ode,func=[x(t),y(t)],ics=ics)
\[
\left [ x{\left (t \right )} = \left (\frac {C_{1}}{2} - \frac {\sqrt {3} C_{2}}{2}\right ) e^{\frac {5 t}{2}} \cos {\left (\frac {\sqrt {3} t}{2} \right )} - \left (\frac {\sqrt {3} C_{1}}{2} + \frac {C_{2}}{2}\right ) e^{\frac {5 t}{2}} \sin {\left (\frac {\sqrt {3} t}{2} \right )}, \ y{\left (t \right )} = C_{1} e^{\frac {5 t}{2}} \cos {\left (\frac {\sqrt {3} t}{2} \right )} - C_{2} e^{\frac {5 t}{2}} \sin {\left (\frac {\sqrt {3} t}{2} \right )}\right ]
\]