64.24.10 problem 10
Internal
problem
ID
[13695]
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
:
Tuesday, January 28, 2025 at 05:55:22 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*}
✓ Solution by Maple
Time used: 0.247 (sec). Leaf size: 245
dsolve([diff(x(t),t)=a*x(t)+b*y(t),diff(y(t),t)=c*x(t)+d*y(t)],singsol=all)
\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*}
✓ Solution by Mathematica
Time used: 0.033 (sec). Leaf size: 410
DSolve[{D[x[t],t]==a*x[t]+b*y[t],D[y[t],t]==c*x[t]+d*y[t]},{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*}