28.4.19 problem 7.19
Internal
problem
ID
[4551]
Book
:
Differential
equations
for
engineers
by
Wei-Chau
XIE,
Cambridge
Press
2010
Section
:
Chapter
7.
Systems
of
linear
differential
equations.
Problems
at
page
351
Problem
number
:
7.19
Date
solved
:
Sunday, March 30, 2025 at 03:26:05 AM
CAS
classification
:
system_of_ODEs
\begin{align*} \frac {d}{d t}x \left (t \right )-x \left (t \right )+y \left (t \right )&=\sec \left (t \right )\\ -2 x \left (t \right )+\frac {d}{d t}y \left (t \right )+y \left (t \right )&=0 \end{align*}
✓ Maple. Time used: 0.269 (sec). Leaf size: 78
ode:=[diff(x(t),t)-x(t)+y(t) = sec(t), -2*x(t)+diff(y(t),t)+y(t) = 0];
dsolve(ode);
\begin{align*}
x \left (t \right ) &= \frac {\cos \left (t \right ) c_2}{2}-\frac {\sin \left (t \right ) c_1}{2}+\sin \left (t \right )+t \cos \left (t \right )-\tan \left (t \right ) \cos \left (t \right )+\ln \left (\sec \left (t \right )\right ) \sin \left (t \right )+\frac {\sin \left (t \right ) c_2}{2}+\frac {\cos \left (t \right ) c_1}{2}+t \sin \left (t \right )-\ln \left (\sec \left (t \right )\right ) \cos \left (t \right ) \\
y \left (t \right ) &= \sin \left (t \right ) c_2 +\cos \left (t \right ) c_1 +2 t \sin \left (t \right )-2 \ln \left (\sec \left (t \right )\right ) \cos \left (t \right ) \\
\end{align*}
✓ Mathematica. Time used: 0.015 (sec). Leaf size: 62
ode={D[x[t],t]-x[t]+y[t]==Sec[t],-2*x[t]+D[y[t],t]+y[t]==0};
ic={};
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
\begin{align*}
x(t)\to \cos (t) (t+\log (\cos (t))+c_1)+\sin (t) (t-\log (\cos (t))+c_1-c_2) \\
y(t)\to (2 t+2 c_1-c_2) \sin (t)+\cos (t) (2 \log (\cos (t))+c_2) \\
\end{align*}
✓ Sympy. Time used: 0.136 (sec). Leaf size: 78
from sympy import *
t = symbols("t")
x = Function("x")
y = Function("y")
ode=[Eq(-x(t) + y(t) + Derivative(x(t), t) - 1/cos(t),0),Eq(-2*x(t) + y(t) + Derivative(y(t), t),0)]
ics = {}
dsolve(ode,func=[x(t),y(t)],ics=ics)
\[
\left [ x{\left (t \right )} = t \sin {\left (t \right )} + t \cos {\left (t \right )} + \left (\frac {C_{1}}{2} - \frac {C_{2}}{2}\right ) \cos {\left (t \right )} - \left (\frac {C_{1}}{2} + \frac {C_{2}}{2}\right ) \sin {\left (t \right )} - \log {\left (\cos {\left (t \right )} \right )} \sin {\left (t \right )} + \log {\left (\cos {\left (t \right )} \right )} \cos {\left (t \right )}, \ y{\left (t \right )} = C_{1} \cos {\left (t \right )} - C_{2} \sin {\left (t \right )} + 2 t \sin {\left (t \right )} + 2 \log {\left (\cos {\left (t \right )} \right )} \cos {\left (t \right )}\right ]
\]