71.19.3 problem 3
Internal
problem
ID
[14524]
Book
:
Ordinary
Differential
Equations
by
Charles
E.
Roberts,
Jr.
CRC
Press.
2010
Section
:
Chapter
10.
Applications
of
Systems
of
Equations.
Exercises
10.2
page
432
Problem
number
:
3
Date
solved
:
Monday, March 31, 2025 at 12:29:30 PM
CAS
classification
:
system_of_ODEs
\begin{align*} \frac {d}{d t}x \left (t \right )&=-x \left (t \right )-2 y \left (t \right )\\ \frac {d}{d t}y \left (t \right )&=2 x \left (t \right )-3 y \left (t \right ) \end{align*}
✓ Maple. Time used: 0.139 (sec). Leaf size: 75
ode:=[diff(x(t),t) = -x(t)-2*y(t), diff(y(t),t) = 2*x(t)-3*y(t)];
dsolve(ode);
\begin{align*}
x \left (t \right ) &= {\mathrm e}^{-2 t} \left (\sin \left (\sqrt {3}\, t \right ) c_1 +\cos \left (\sqrt {3}\, t \right ) c_2 \right ) \\
y \left (t \right ) &= \frac {{\mathrm e}^{-2 t} \left (\sin \left (\sqrt {3}\, t \right ) \sqrt {3}\, c_2 -\cos \left (\sqrt {3}\, t \right ) \sqrt {3}\, c_1 +\sin \left (\sqrt {3}\, t \right ) c_1 +\cos \left (\sqrt {3}\, t \right ) c_2 \right )}{2} \\
\end{align*}
✓ Mathematica. Time used: 0.019 (sec). Leaf size: 96
ode={D[x[t],t]==-x[t]-2*y[t],D[y[t],t]==2*x[t]-3*y[t]};
ic={};
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
\begin{align*}
x(t)\to \frac {1}{3} e^{-2 t} \left (3 c_1 \cos \left (\sqrt {3} t\right )+\sqrt {3} (c_1-2 c_2) \sin \left (\sqrt {3} t\right )\right ) \\
y(t)\to \frac {1}{3} e^{-2 t} \left (3 c_2 \cos \left (\sqrt {3} t\right )+\sqrt {3} (2 c_1-c_2) \sin \left (\sqrt {3} t\right )\right ) \\
\end{align*}
✓ Sympy. Time used: 0.185 (sec). Leaf size: 85
from sympy import *
t = symbols("t")
x = Function("x")
y = Function("y")
ode=[Eq(x(t) + 2*y(t) + Derivative(x(t), t),0),Eq(-2*x(t) + 3*y(t) + Derivative(y(t), t),0)]
ics = {}
dsolve(ode,func=[x(t),y(t)],ics=ics)
\[
\left [ x{\left (t \right )} = \left (\frac {C_{1}}{2} - \frac {\sqrt {3} C_{2}}{2}\right ) e^{- 2 t} \cos {\left (\sqrt {3} t \right )} - \left (\frac {\sqrt {3} C_{1}}{2} + \frac {C_{2}}{2}\right ) e^{- 2 t} \sin {\left (\sqrt {3} t \right )}, \ y{\left (t \right )} = C_{1} e^{- 2 t} \cos {\left (\sqrt {3} t \right )} - C_{2} e^{- 2 t} \sin {\left (\sqrt {3} t \right )}\right ]
\]