72.9.8 problem 8
Internal
problem
ID
[14733]
Book
:
DIFFERENTIAL
EQUATIONS
by
Paul
Blanchard,
Robert
L.
Devaney,
Glen
R.
Hall.
4th
edition.
Brooks/Cole.
Boston,
USA.
2012
Section
:
Chapter
3.
Linear
Systems.
Exercises
section
3.1.
page
258
Problem
number
:
8
Date
solved
:
Monday, March 31, 2025 at 12:57:01 PM
CAS
classification
:
system_of_ODEs
\begin{align*} \frac {d}{d t}x \left (t \right )&=-3 x \left (t \right )+2 \pi y \left (t \right )\\ \frac {d}{d t}y \left (t \right )&=4 x \left (t \right )-y \left (t \right ) \end{align*}
✓ Maple. Time used: 0.105 (sec). Leaf size: 118
ode:=[diff(x(t),t) = -3*x(t)+2*Pi*y(t), diff(y(t),t) = 4*x(t)-y(t)];
dsolve(ode);
\begin{align*}
x \left (t \right ) &= c_1 \,{\mathrm e}^{-\left (2+\sqrt {1+8 \pi }\right ) t}+c_2 \,{\mathrm e}^{\left (-2+\sqrt {1+8 \pi }\right ) t} \\
y \left (t \right ) &= -\frac {c_1 \,{\mathrm e}^{-\left (2+\sqrt {1+8 \pi }\right ) t} \sqrt {1+8 \pi }-c_2 \,{\mathrm e}^{\left (-2+\sqrt {1+8 \pi }\right ) t} \sqrt {1+8 \pi }-c_1 \,{\mathrm e}^{-\left (2+\sqrt {1+8 \pi }\right ) t}-c_2 \,{\mathrm e}^{\left (-2+\sqrt {1+8 \pi }\right ) t}}{2 \pi } \\
\end{align*}
✓ Mathematica. Time used: 0.007 (sec). Leaf size: 189
ode={D[x[t],t]==-3*x[t]+2*Pi*y[t],D[y[t],t]==4*x[t]-y[t]};
ic={};
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
\begin{align*}
x(t)\to \frac {e^{-\left (\left (2+\sqrt {1+8 \pi }\right ) t\right )} \left (c_1 \left (\left (\sqrt {1+8 \pi }-1\right ) e^{2 \sqrt {1+8 \pi } t}+1+\sqrt {1+8 \pi }\right )+2 \pi c_2 \left (e^{2 \sqrt {1+8 \pi } t}-1\right )\right )}{2 \sqrt {1+8 \pi }} \\
y(t)\to \frac {e^{-\left (\left (2+\sqrt {1+8 \pi }\right ) t\right )} \left (4 c_1 \left (e^{2 \sqrt {1+8 \pi } t}-1\right )+c_2 \left (\left (1+\sqrt {1+8 \pi }\right ) e^{2 \sqrt {1+8 \pi } t}-1+\sqrt {1+8 \pi }\right )\right )}{2 \sqrt {1+8 \pi }} \\
\end{align*}
✓ Sympy. Time used: 0.226 (sec). Leaf size: 90
from sympy import *
t = symbols("t")
x = Function("x")
y = Function("y")
ode=[Eq(3*x(t) - 2*pi*y(t) + Derivative(x(t), t),0),Eq(-4*x(t) + y(t) + Derivative(y(t), t),0)]
ics = {}
dsolve(ode,func=[x(t),y(t)],ics=ics)
\[
\left [ x{\left (t \right )} = - \frac {C_{1} \left (1 - \sqrt {1 + 8 \pi }\right ) e^{- t \left (2 - \sqrt {1 + 8 \pi }\right )}}{4} - \frac {C_{2} \left (1 + \sqrt {1 + 8 \pi }\right ) e^{- t \left (2 + \sqrt {1 + 8 \pi }\right )}}{4}, \ y{\left (t \right )} = C_{1} e^{- t \left (2 - \sqrt {1 + 8 \pi }\right )} + C_{2} e^{- t \left (2 + \sqrt {1 + 8 \pi }\right )}\right ]
\]