67.28.2 problem 39.1 (b)
Internal
problem
ID
[17064]
Book
:
Ordinary
Differential
Equations.
An
introduction
to
the
fundamentals.
Kenneth
B.
Howell.
second
edition.
CRC
Press.
FL,
USA.
2020
Section
:
Chapter
39.
Critical
points,
Direction
fields
and
trajectories.
Additional
Exercises.
page
815
Problem
number
:
39.1
(b)
Date
solved
:
Thursday, October 02, 2025 at 01:42:31 PM
CAS
classification
:
system_of_ODEs
\begin{align*} \frac {d}{d t}x \left (t \right )&=2 x \left (t \right )-5 y \left (t \right )+4\\ \frac {d}{d t}y \left (t \right )&=3 x \left (t \right )-7 y \left (t \right )+5 \end{align*}
✓ Maple. Time used: 0.130 (sec). Leaf size: 87
ode:=[diff(x(t),t) = 2*x(t)-5*y(t)+4, diff(y(t),t) = 3*x(t)-7*y(t)+5];
dsolve(ode);
\begin{align*}
x \left (t \right ) &= {\mathrm e}^{\frac {\left (-5+\sqrt {21}\right ) t}{2}} c_2 +{\mathrm e}^{-\frac {\left (5+\sqrt {21}\right ) t}{2}} c_1 +3 \\
y \left (t \right ) &= -\frac {{\mathrm e}^{\frac {\left (-5+\sqrt {21}\right ) t}{2}} c_2 \sqrt {21}}{10}+\frac {{\mathrm e}^{-\frac {\left (5+\sqrt {21}\right ) t}{2}} c_1 \sqrt {21}}{10}+\frac {9 \,{\mathrm e}^{\frac {\left (-5+\sqrt {21}\right ) t}{2}} c_2}{10}+\frac {9 \,{\mathrm e}^{-\frac {\left (5+\sqrt {21}\right ) t}{2}} c_1}{10}+2 \\
\end{align*}
✓ Mathematica. Time used: 0.442 (sec). Leaf size: 185
ode={D[x[t],t]==2*x[t]-5*y[t]+4,D[y[t],t]==3*x[t]-7*y[t]+5};
ic={};
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
\begin{align*} x(t)&\to \frac {1}{42} e^{-\frac {1}{2} \left (5+\sqrt {21}\right ) t} \left (126 e^{\frac {1}{2} \left (5+\sqrt {21}\right ) t}+\left (3 \left (7+3 \sqrt {21}\right ) c_1-10 \sqrt {21} c_2\right ) e^{\sqrt {21} t}+\left (21-9 \sqrt {21}\right ) c_1+10 \sqrt {21} c_2\right )\\ y(t)&\to \frac {1}{14} e^{-\frac {1}{2} \left (5+\sqrt {21}\right ) t} \left (28 e^{\frac {1}{2} \left (5+\sqrt {21}\right ) t}+\left (2 \sqrt {21} c_1+\left (7-3 \sqrt {21}\right ) c_2\right ) e^{\sqrt {21} t}-2 \sqrt {21} c_1+\left (7+3 \sqrt {21}\right ) c_2\right ) \end{align*}
✓ Sympy. Time used: 0.269 (sec). Leaf size: 78
from sympy import *
t = symbols("t")
x = Function("x")
y = Function("y")
ode=[Eq(-2*x(t) + 5*y(t) + Derivative(x(t), t) - 4,0),Eq(-3*x(t) + 7*y(t) + Derivative(y(t), t) - 5,0)]
ics = {}
dsolve(ode,func=[x(t),y(t)],ics=ics)
\[
\left [ x{\left (t \right )} = \frac {C_{1} \left (9 - \sqrt {21}\right ) e^{- \frac {t \left (\sqrt {21} + 5\right )}{2}}}{6} + \frac {C_{2} \left (\sqrt {21} + 9\right ) e^{- \frac {t \left (5 - \sqrt {21}\right )}{2}}}{6} + 3, \ y{\left (t \right )} = C_{1} e^{- \frac {t \left (\sqrt {21} + 5\right )}{2}} + C_{2} e^{- \frac {t \left (5 - \sqrt {21}\right )}{2}} + 2\right ]
\]