7.23.11 problem 11
Internal
problem
ID
[597]
Book
:
Elementary
Differential
Equations.
By
C.
Henry
Edwards,
David
E.
Penney
and
David
Calvis.
6th
edition.
2008
Section
:
Chapter
5.
Linear
systems
of
differential
equations.
Section
5.2
(Applications).
Problems
at
page
345
Problem
number
:
11
Date
solved
:
Saturday, March 29, 2025 at 04:57:36 PM
CAS
classification
:
system_of_ODEs
\begin{align*} -\frac {d}{d t}x \left (t \right )+2 \frac {d}{d t}y \left (t \right )&=x \left (t \right )+3 y \left (t \right )+{\mathrm e}^{t}\\ 3 \frac {d}{d t}x \left (t \right )-4 \frac {d}{d t}y \left (t \right )&=x \left (t \right )-15 y \left (t \right )+{\mathrm e}^{-t} \end{align*}
✓ Maple. Time used: 0.295 (sec). Leaf size: 63
ode:=[-diff(x(t),t)+2*diff(y(t),t) = x(t)+3*y(t)+exp(t), 3*diff(x(t),t)-4*diff(y(t),t) = x(t)-15*y(t)+exp(-t)];
dsolve(ode);
\begin{align*}
x \left (t \right ) &= \sin \left (3 t \right ) c_2 +\cos \left (3 t \right ) c_1 -\frac {11 \,{\mathrm e}^{t}}{20}-\frac {{\mathrm e}^{-t}}{4} \\
y \left (t \right ) &= \frac {\cos \left (3 t \right ) c_1}{3}-\frac {\cos \left (3 t \right ) c_2}{3}+\frac {\sin \left (3 t \right ) c_1}{3}+\frac {\sin \left (3 t \right ) c_2}{3}+\frac {{\mathrm e}^{t}}{10} \\
\end{align*}
✓ Mathematica. Time used: 0.529 (sec). Leaf size: 77
ode={-D[x[t],t]+2*D[y[t],t]==x[t]+3*y[t]+Exp[t],3*D[x[t],t]-4*D[y[t],t]==x[t]-15*y[t]+Exp[-t]};
ic={};
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
\begin{align*}
x(t)\to -\frac {1}{20} e^{-t} \left (11 e^{2 t}+5\right )+c_1 \cos (3 t)+(c_1-3 c_2) \sin (3 t) \\
y(t)\to \frac {e^t}{10}+c_2 \cos (3 t)+\left (\frac {2 c_1}{3}-c_2\right ) \sin (3 t) \\
\end{align*}
✓ Sympy. Time used: 0.440 (sec). Leaf size: 122
from sympy import *
t = symbols("t")
x = Function("x")
y = Function("y")
ode=[Eq(-x(t) - 3*y(t) - exp(t) - Derivative(x(t), t) + 2*Derivative(y(t), t),0),Eq(-x(t) + 15*y(t) + 3*Derivative(x(t), t) - 4*Derivative(y(t), t) - exp(-t),0)]
ics = {}
dsolve(ode,func=[x(t),y(t)],ics=ics)
\[
\left [ x{\left (t \right )} = \left (\frac {3 C_{1}}{2} - \frac {3 C_{2}}{2}\right ) \cos {\left (3 t \right )} - \left (\frac {3 C_{1}}{2} + \frac {3 C_{2}}{2}\right ) \sin {\left (3 t \right )} - \frac {11 e^{t} \sin ^{2}{\left (3 t \right )}}{20} - \frac {11 e^{t} \cos ^{2}{\left (3 t \right )}}{20} - \frac {e^{- t} \sin ^{2}{\left (3 t \right )}}{4} - \frac {e^{- t} \cos ^{2}{\left (3 t \right )}}{4}, \ y{\left (t \right )} = C_{1} \cos {\left (3 t \right )} - C_{2} \sin {\left (3 t \right )} + \frac {e^{t} \sin ^{2}{\left (3 t \right )}}{10} + \frac {e^{t} \cos ^{2}{\left (3 t \right )}}{10}\right ]
\]