72.10.8 problem 8
Internal
problem
ID
[14740]
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
3.2.
page
277
Problem
number
:
8
Date
solved
:
Thursday, March 13, 2025 at 04:18:01 AM
CAS
classification
:
system_of_ODEs
\begin{align*} \frac {d}{d t}x \left (t \right )&=2 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: 0.040 (sec). Leaf size: 85
ode:=[diff(x(t),t) = 2*x(t)-y(t), diff(y(t),t) = -x(t)+y(t)];
dsolve(ode);
\begin{align*}
x \left (t \right ) &= c_{1} {\mathrm e}^{\frac {\left (\sqrt {5}+3\right ) t}{2}}+c_{2} {\mathrm e}^{-\frac {\left (\sqrt {5}-3\right ) t}{2}} \\
y &= -\frac {c_{1} {\mathrm e}^{\frac {\left (\sqrt {5}+3\right ) t}{2}} \sqrt {5}}{2}+\frac {c_{2} {\mathrm e}^{-\frac {\left (\sqrt {5}-3\right ) t}{2}} \sqrt {5}}{2}+\frac {c_{1} {\mathrm e}^{\frac {\left (\sqrt {5}+3\right ) t}{2}}}{2}+\frac {c_{2} {\mathrm e}^{-\frac {\left (\sqrt {5}-3\right ) t}{2}}}{2} \\
\end{align*}
✓ Mathematica. Time used: 0.008 (sec). Leaf size: 144
ode={D[x[t],t]==2*x[t]-y[t],D[y[t],t]==-1*x[t]+y[t]};
ic={};
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
\begin{align*}
x(t)\to \frac {1}{10} e^{-\frac {1}{2} \left (\sqrt {5}-3\right ) t} \left (c_1 \left (\left (5+\sqrt {5}\right ) e^{\sqrt {5} t}+5-\sqrt {5}\right )-2 \sqrt {5} c_2 \left (e^{\sqrt {5} t}-1\right )\right ) \\
y(t)\to -\frac {1}{10} e^{-\frac {1}{2} \left (\sqrt {5}-3\right ) t} \left (2 \sqrt {5} c_1 \left (e^{\sqrt {5} t}-1\right )+c_2 \left (\left (\sqrt {5}-5\right ) e^{\sqrt {5} t}-5-\sqrt {5}\right )\right ) \\
\end{align*}
✓ Sympy. Time used: 0.187 (sec). Leaf size: 76
from sympy import *
t = symbols("t")
x = Function("x")
y = Function("y")
ode=[Eq(-2*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)
\[
\left [ x{\left (t \right )} = - \frac {C_{1} \left (1 - \sqrt {5}\right ) e^{\frac {t \left (3 - \sqrt {5}\right )}{2}}}{2} - \frac {C_{2} \left (1 + \sqrt {5}\right ) e^{\frac {t \left (\sqrt {5} + 3\right )}{2}}}{2}, \ y{\left (t \right )} = C_{1} e^{\frac {t \left (3 - \sqrt {5}\right )}{2}} + C_{2} e^{\frac {t \left (\sqrt {5} + 3\right )}{2}}\right ]
\]