64.24.10 problem 10
Internal
problem
ID
[13624]
Book
:
Differential
Equations
by
Shepley
L.
Ross.
Third
edition.
John
Willey.
New
Delhi.
2004.
Section
:
Chapter
13,
Nonlinear
differential
equations.
Section
13.2,
Exercises
page
656
Problem
number
:
10
Date
solved
:
Monday, March 31, 2025 at 08:03:01 AM
CAS
classification
:
system_of_ODEs
\begin{align*} \frac {d}{d t}x \left (t \right )&=a x \left (t \right )+b y \left (t \right )\\ \frac {d}{d t}y \left (t \right )&=c x \left (t \right )+d y \left (t \right ) \end{align*}
✓ Maple. Time used: 0.125 (sec). Leaf size: 245
ode:=[diff(x(t),t) = a*x(t)+b*y(t), diff(y(t),t) = c*x(t)+d*y(t)];
dsolve(ode);
\begin{align*}
x \left (t \right ) &= c_1 \,{\mathrm e}^{\frac {\left (a +d +\sqrt {a^{2}-2 a d +4 b c +d^{2}}\right ) t}{2}}+c_2 \,{\mathrm e}^{-\frac {\left (-a -d +\sqrt {a^{2}-2 a d +4 b c +d^{2}}\right ) t}{2}} \\
y \left (t \right ) &= \left (\frac {d}{2 b}+\frac {\frac {\sqrt {a^{2}-2 a d +4 b c +d^{2}}}{2}-\frac {a}{2}}{b}\right ) c_1 \,{\mathrm e}^{\frac {\left (a +d +\sqrt {a^{2}-2 a d +4 b c +d^{2}}\right ) t}{2}}+\left (\frac {{\mathrm e}^{-\frac {\left (-a -d +\sqrt {a^{2}-2 a d +4 b c +d^{2}}\right ) t}{2}} d}{2 b}+\frac {-\frac {{\mathrm e}^{-\frac {\left (-a -d +\sqrt {a^{2}-2 a d +4 b c +d^{2}}\right ) t}{2}} \sqrt {a^{2}-2 a d +4 b c +d^{2}}}{2}-\frac {{\mathrm e}^{-\frac {\left (-a -d +\sqrt {a^{2}-2 a d +4 b c +d^{2}}\right ) t}{2}} a}{2}}{b}\right ) c_2 \\
\end{align*}
✓ Mathematica. Time used: 0.033 (sec). Leaf size: 410
ode={D[x[t],t]==a*x[t]+b*y[t],D[y[t],t]==c*x[t]+d*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 {a^2-2 a d+4 b c+d^2}+a+d\right )} \left (c_1 \sqrt {a^2-2 a d+4 b c+d^2} e^{t \sqrt {a^2-2 a d+4 b c+d^2}}+a c_1 \left (e^{t \sqrt {a^2-2 a d+4 b c+d^2}}-1\right )-c_1 d \left (e^{t \sqrt {a^2-2 a d+4 b c+d^2}}-1\right )+2 b c_2 e^{t \sqrt {a^2-2 a d+4 b c+d^2}}+c_1 \sqrt {a^2-2 a d+4 b c+d^2}-2 b c_2\right )}{2 \sqrt {a^2-2 a d+4 b c+d^2}} \\
y(t)\to \frac {e^{\frac {1}{2} t \left (-\sqrt {a^2-2 a d+4 b c+d^2}+a+d\right )} \left (2 c c_1 \left (e^{t \sqrt {a^2-2 a d+4 b c+d^2}}-1\right )+c_2 \left (a \left (-e^{t \sqrt {a^2-2 a d+4 b c+d^2}}\right )+d \left (e^{t \sqrt {a^2-2 a d+4 b c+d^2}}-1\right )+\sqrt {a^2-2 a d+4 b c+d^2} \left (e^{t \sqrt {a^2-2 a d+4 b c+d^2}}+1\right )+a\right )\right )}{2 \sqrt {a^2-2 a d+4 b c+d^2}} \\
\end{align*}
✓ Sympy. Time used: 0.282 (sec). Leaf size: 180
from sympy import *
t = symbols("t")
a = symbols("a")
b = symbols("b")
c = symbols("c")
d = symbols("d")
x = Function("x")
y = Function("y")
ode=[Eq(-a*x(t) - b*y(t) + Derivative(x(t), t),0),Eq(-c*x(t) - d*y(t) + Derivative(y(t), t),0)]
ics = {}
dsolve(ode,func=[x(t),y(t)],ics=ics)
\[
\left [ x{\left (t \right )} = \frac {2 C_{1} b e^{\frac {t \left (a + d + \sqrt {a^{2} - 2 a d + 4 b c + d^{2}}\right )}{2}}}{- a + d + \sqrt {a^{2} - 2 a d + 4 b c + d^{2}}} - \frac {2 C_{2} b e^{\frac {t \left (a + d - \sqrt {a^{2} - 2 a d + 4 b c + d^{2}}\right )}{2}}}{a - d + \sqrt {a^{2} - 2 a d + 4 b c + d^{2}}}, \ y{\left (t \right )} = C_{1} e^{\frac {t \left (a + d + \sqrt {a^{2} - 2 a d + 4 b c + d^{2}}\right )}{2}} + C_{2} e^{\frac {t \left (a + d - \sqrt {a^{2} - 2 a d + 4 b c + d^{2}}\right )}{2}}\right ]
\]