14.20.11 problem 11
Internal
problem
ID
[2708]
Book
:
Differential
equations
and
their
applications,
4th
ed.,
M.
Braun
Section
:
Chapter
2.
Second
order
differential
equations.
Section
2.14,
The
method
of
elimination
for
systems.
Excercises
page
258
Problem
number
:
11
Date
solved
:
Monday, January 27, 2025 at 06:12:07 AM
CAS
classification
:
system_of_ODEs
\begin{align*} x^{\prime }\left (t \right )&=2 x \left (t \right )-5 y+\sin \left (t \right )\\ y^{\prime }&=x \left (t \right )-2 y+\tan \left (t \right ) \end{align*}
With initial conditions
\begin{align*} x \left (0\right ) = 0\\ y \left (0\right ) = 0 \end{align*}
✓ Solution by Maple
Time used: 0.327 (sec). Leaf size: 148
dsolve([diff(x(t),t) = 2*x(t)-5*y(t)+sin(t), diff(y(t),t) = x(t)-2*y(t)+tan(t), x(0) = 0, y(0) = 0], singsol=all)
\begin{align*}
x \left (t \right ) &= -4 \sin \left (t \right )+\frac {\sin \left (t \right ) t}{2}+5 \cos \left (t \right ) \ln \left (\sec \left (t \right )+\tan \left (t \right )\right )-\cos \left (t \right ) t \\
y &= -\frac {10 \tan \left (t \right )^{2} \cos \left (t \right )+10 \tan \left (t \right ) \sec \left (t \right ) \cos \left (t \right )-20 \cos \left (t \right ) \ln \left (\sec \left (t \right )+\tan \left (t \right )\right ) \tan \left (t \right )-10 \sin \left (t \right ) \ln \left (\sec \left (t \right )+\tan \left (t \right )\right ) \tan \left (t \right )-10 \cos \left (t \right ) \tan \left (t \right )+5 \cos \left (t \right ) t \tan \left (t \right )+15 \sin \left (t \right ) \tan \left (t \right )-20 \cos \left (t \right ) \ln \left (\sec \left (t \right )+\tan \left (t \right )\right ) \sec \left (t \right )-10 \sin \left (t \right ) \ln \left (\sec \left (t \right )+\tan \left (t \right )\right ) \sec \left (t \right )-10 \cos \left (t \right ) \sec \left (t \right )+5 \cos \left (t \right ) t \sec \left (t \right )+15 \sin \left (t \right ) \sec \left (t \right )+10 \cos \left (t \right )}{10 \left (\sec \left (t \right )+\tan \left (t \right )\right )} \\
\end{align*}
✓ Solution by Mathematica
Time used: 0.074 (sec). Leaf size: 58
DSolve[{D[x[t],t]==2*x[t]-5*y[t]+Sin[t],D[y[t],t]==x[t]-2*y[t]+Tan[t]},{x[0]==0,y[0]==0},{x[t],y[t]},t,IncludeSingularSolutions -> True]
\begin{align*}
x(t)\to 5 \cos (t) \text {arctanh}(\sin (t))+\frac {1}{2} (t-8) \sin (t)-t \cos (t) \\
y(t)\to \text {arctanh}(\sin (t)) (\sin (t)+2 \cos (t))-\frac {3 \sin (t)}{2}-\frac {1}{2} t \cos (t)+\cos (t)-1 \\
\end{align*}