72.15.5 problem 19 (ii)
Internal
problem
ID
[14807]
Book
:
DIFFERENTIAL
EQUATIONS
by
Paul
Blanchard,
Robert
L.
Devaney,
Glen
R.
Hall.
4th
edition.
Brooks/Cole.
Boston,
USA.
2012
Section
:
Chapter
3.
Linear
Systems.
Review
Exercises
for
chapter
3.
page
376
Problem
number
:
19
(ii)
Date
solved
:
Thursday, March 13, 2025 at 04:19:19 AM
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 )+y \left (t \right ) \end{align*}
✓ Maple. Time used: 0.050 (sec). Leaf size: 81
ode:=[diff(x(t),t) = -3*x(t)+y(t), diff(y(t),t) = -x(t)+y(t)];
dsolve(ode);
\begin{align*}
x \left (t \right ) &= c_{1} {\mathrm e}^{\left (\sqrt {3}-1\right ) t}+c_{2} {\mathrm e}^{-\left (1+\sqrt {3}\right ) t} \\
y &= c_{1} {\mathrm e}^{\left (\sqrt {3}-1\right ) t} \sqrt {3}-c_{2} {\mathrm e}^{-\left (1+\sqrt {3}\right ) t} \sqrt {3}+2 c_{1} {\mathrm e}^{\left (\sqrt {3}-1\right ) t}+2 c_{2} {\mathrm e}^{-\left (1+\sqrt {3}\right ) t} \\
\end{align*}
✓ Mathematica. Time used: 0.008 (sec). Leaf size: 147
ode={D[x[t],t]==-3*x[t]+1*y[t],D[y[t],t]==-1*x[t]+1*y[t]};
ic={};
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
\begin{align*}
x(t)\to \frac {1}{6} e^{-\left (\left (1+\sqrt {3}\right ) t\right )} \left (c_1 \left (\left (3-2 \sqrt {3}\right ) e^{2 \sqrt {3} t}+3+2 \sqrt {3}\right )+\sqrt {3} c_2 \left (e^{2 \sqrt {3} t}-1\right )\right ) \\
y(t)\to \frac {1}{6} e^{-\left (\left (1+\sqrt {3}\right ) t\right )} \left (c_2 \left (\left (3+2 \sqrt {3}\right ) e^{2 \sqrt {3} t}+3-2 \sqrt {3}\right )-\sqrt {3} c_1 \left (e^{2 \sqrt {3} t}-1\right )\right ) \\
\end{align*}
✓ Sympy. Time used: 0.190 (sec). Leaf size: 65
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) - y(t) + Derivative(y(t), t),0)]
ics = {}
dsolve(ode,func=[x(t),y(t)],ics=ics)
\[
\left [ x{\left (t \right )} = C_{1} \left (2 - \sqrt {3}\right ) e^{- t \left (1 - \sqrt {3}\right )} + C_{2} \left (\sqrt {3} + 2\right ) e^{- t \left (1 + \sqrt {3}\right )}, \ y{\left (t \right )} = C_{1} e^{- t \left (1 - \sqrt {3}\right )} + C_{2} e^{- t \left (1 + \sqrt {3}\right )}\right ]
\]