72.9.9 problem 9
Internal
problem
ID
[14726]
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
:
9
Date
solved
:
Thursday, March 13, 2025 at 04:17:46 AM
CAS
classification
:
system_of_ODEs
\begin{align*} \frac {d}{d t}x \left (t \right )&=\beta y \left (t \right )\\ \frac {d}{d t}y \left (t \right )&=\gamma x \left (t \right )-y \left (t \right ) \end{align*}
✓ Maple. Time used: 0.125 (sec). Leaf size: 118
ode:=[diff(x(t),t) = beta*y(t), diff(y(t),t) = gamma*x(t)-y(t)];
dsolve(ode);
\begin{align*}
x \left (t \right ) &= c_{1} {\mathrm e}^{\frac {\left (-1+\sqrt {4 \beta \gamma +1}\right ) t}{2}}+c_{2} {\mathrm e}^{-\frac {\left (1+\sqrt {4 \beta \gamma +1}\right ) t}{2}} \\
y &= \frac {\left (-\frac {1}{2}+\frac {\sqrt {4 \beta \gamma +1}}{2}\right ) c_{1} {\mathrm e}^{\frac {\left (-1+\sqrt {4 \beta \gamma +1}\right ) t}{2}}}{\beta }+\frac {\left (-\frac {{\mathrm e}^{-\frac {\left (1+\sqrt {4 \beta \gamma +1}\right ) t}{2}} \sqrt {4 \beta \gamma +1}}{2}-\frac {{\mathrm e}^{-\frac {\left (1+\sqrt {4 \beta \gamma +1}\right ) t}{2}}}{2}\right ) c_{2}}{\beta } \\
\end{align*}
✓ Mathematica. Time used: 0.01 (sec). Leaf size: 202
ode={D[x[t],t]==\[Beta]*y[t],D[y[t],t]==\[Gamma]*x[t]-y[t]};
ic={};
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
\begin{align*}
x(t)\to \frac {e^{-\frac {1}{2} t \left (\sqrt {4 \beta \gamma +1}+1\right )} \left (c_1 \left (\sqrt {4 \beta \gamma +1}+\left (\sqrt {4 \beta \gamma +1}+1\right ) e^{t \sqrt {4 \beta \gamma +1}}-1\right )+2 \beta c_2 \left (e^{t \sqrt {4 \beta \gamma +1}}-1\right )\right )}{2 \sqrt {4 \beta \gamma +1}} \\
y(t)\to \frac {e^{-\frac {1}{2} t \left (\sqrt {4 \beta \gamma +1}+1\right )} \left (2 \gamma c_1 \left (e^{t \sqrt {4 \beta \gamma +1}}-1\right )+c_2 \left (\sqrt {4 \beta \gamma +1}+\left (\sqrt {4 \beta \gamma +1}-1\right ) e^{t \sqrt {4 \beta \gamma +1}}+1\right )\right )}{2 \sqrt {4 \beta \gamma +1}} \\
\end{align*}
✓ Sympy. Time used: 0.251 (sec). Leaf size: 109
from sympy import *
t = symbols("t")
BETA = symbols("BETA")
Gamma = symbols("Gamma")
x = Function("x")
y = Function("y")
ode=[Eq(-BETA*y(t) + Derivative(x(t), t),0),Eq(-Gamma*x(t) + y(t) + Derivative(y(t), t),0)]
ics = {}
dsolve(ode,func=[x(t),y(t)],ics=ics)
\[
\left [ x{\left (t \right )} = \frac {2 \beta C_{1} e^{\frac {t \left (\sqrt {4 \beta \Gamma + 1} - 1\right )}{2}}}{\sqrt {4 \beta \Gamma + 1} - 1} - \frac {2 \beta C_{2} e^{- \frac {t \left (\sqrt {4 \beta \Gamma + 1} + 1\right )}{2}}}{\sqrt {4 \beta \Gamma + 1} + 1}, \ y{\left (t \right )} = C_{1} e^{\frac {t \left (\sqrt {4 \beta \Gamma + 1} - 1\right )}{2}} + C_{2} e^{- \frac {t \left (\sqrt {4 \beta \Gamma + 1} + 1\right )}{2}}\right ]
\]