50.29.7 problem 3(c)
Internal
problem
ID
[8203]
Book
:
Differential
Equations:
Theory,
Technique,
and
Practice
by
George
Simmons,
Steven
Krantz.
McGraw-Hill
NY.
2007.
1st
Edition.
Section
:
Chapter
10.
Systems
of
First-Order
Equations.
Section
A.
Drill
exercises.
Page
400
Problem
number
:
3(c)
Date
solved
:
Wednesday, March 05, 2025 at 05:32:01 AM
CAS
classification
:
system_of_ODEs
\begin{align*} \frac {d}{d t}x \left (t \right )&=3 x \left (t \right )-5 y \left (t \right )\\ \frac {d}{d t}y \left (t \right )&=-x \left (t \right )+2 y \left (t \right ) \end{align*}
✓ Maple. Time used: 0.027 (sec). Leaf size: 85
ode:=[diff(x(t),t) = 3*x(t)-5*y(t), diff(y(t),t) = -x(t)+2*y(t)];
dsolve(ode);
\begin{align*}
x \left (t \right ) &= c_{1} {\mathrm e}^{\frac {\left (5+\sqrt {21}\right ) t}{2}}+c_{2} {\mathrm e}^{-\frac {\left (-5+\sqrt {21}\right ) t}{2}} \\
y &= -\frac {c_{1} {\mathrm e}^{\frac {\left (5+\sqrt {21}\right ) t}{2}} \sqrt {21}}{10}+\frac {c_{2} {\mathrm e}^{-\frac {\left (-5+\sqrt {21}\right ) t}{2}} \sqrt {21}}{10}+\frac {c_{1} {\mathrm e}^{\frac {\left (5+\sqrt {21}\right ) t}{2}}}{10}+\frac {c_{2} {\mathrm e}^{-\frac {\left (-5+\sqrt {21}\right ) t}{2}}}{10} \\
\end{align*}
✓ Mathematica. Time used: 0.011 (sec). Leaf size: 144
ode={D[x[t],t]==3*x[t]-5*y[t],D[y[t],t]==-x[t]+2*y[t]};
ic={};
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
\begin{align*}
x(t)\to \frac {1}{42} e^{-\frac {1}{2} \left (\sqrt {21}-5\right ) t} \left (c_1 \left (\left (21+\sqrt {21}\right ) e^{\sqrt {21} t}+21-\sqrt {21}\right )-10 \sqrt {21} c_2 \left (e^{\sqrt {21} t}-1\right )\right ) \\
y(t)\to -\frac {1}{42} e^{-\frac {1}{2} \left (\sqrt {21}-5\right ) t} \left (2 \sqrt {21} c_1 \left (e^{\sqrt {21} t}-1\right )+c_2 \left (\left (\sqrt {21}-21\right ) e^{\sqrt {21} t}-21-\sqrt {21}\right )\right ) \\
\end{align*}
✓ Sympy. Time used: 0.190 (sec). Leaf size: 76
from sympy import *
t = symbols("t")
x = Function("x")
y = Function("y")
ode=[Eq(-3*x(t) + 5*y(t) + Derivative(x(t), t),0),Eq(x(t) - 2*y(t) + Derivative(y(t), t),0)]
ics = {}
dsolve(ode,func=[x(t),y(t)],ics=ics)
\[
\left [ x{\left (t \right )} = - \frac {C_{1} \left (1 - \sqrt {21}\right ) e^{\frac {t \left (5 - \sqrt {21}\right )}{2}}}{2} - \frac {C_{2} \left (1 + \sqrt {21}\right ) e^{\frac {t \left (\sqrt {21} + 5\right )}{2}}}{2}, \ y{\left (t \right )} = C_{1} e^{\frac {t \left (5 - \sqrt {21}\right )}{2}} + C_{2} e^{\frac {t \left (\sqrt {21} + 5\right )}{2}}\right ]
\]