14.26.6 problem 6
Internal
problem
ID
[2779]
Book
:
Differential
equations
and
their
applications,
4th
ed.,
M.
Braun
Section
:
Chapter
3.
Systems
of
differential
equations.
Section
3.13
(Solving
systems
by
Laplace
transform).
Page
370
Problem
number
:
6
Date
solved
:
Monday, January 27, 2025 at 06:13:13 AM
CAS
classification
:
system_of_ODEs
\begin{align*} x_{1}^{\prime }\left (t \right )&=2 x_{1} \left (t \right )-5 x_{2} \left (t \right )+\sin \left (t \right )\\ x_{2}^{\prime }\left (t \right )&=x_{1} \left (t \right )-2 x_{2} \left (t \right )+\tan \left (t \right ) \end{align*}
With initial conditions
\begin{align*} x_{1} \left (0\right ) = -1\\ x_{2} \left (0\right ) = 0 \end{align*}
✓ Solution by Maple
Time used: 0.059 (sec). Leaf size: 152
dsolve([diff(x__1(t),t) = 2*x__1(t)-5*x__2(t)+sin(t), diff(x__2(t),t) = x__1(t)-2*x__2(t)+tan(t), x__1(0) = -1, x__2(0) = 0], singsol=all)
\begin{align*}
x_{1} \left (t \right ) &= -6 \sin \left (t \right )-\cos \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 \\
x_{2} \left (t \right ) &= -\frac {10 \tan \left (t \right )^{2} \cos \left (t \right )+10 \tan \left (t \right ) \sec \left (t \right ) \cos \left (t \right )-10 \sin \left (t \right ) \ln \left (\sec \left (t \right )+\tan \left (t \right )\right ) \tan \left (t \right )-20 \cos \left (t \right ) \ln \left (\sec \left (t \right )+\tan \left (t \right )\right ) \tan \left (t \right )+25 \sin \left (t \right ) \tan \left (t \right )-10 \cos \left (t \right ) \tan \left (t \right )+5 \cos \left (t \right ) t \tan \left (t \right )-10 \sin \left (t \right ) \ln \left (\sec \left (t \right )+\tan \left (t \right )\right ) \sec \left (t \right )-20 \cos \left (t \right ) \ln \left (\sec \left (t \right )+\tan \left (t \right )\right ) \sec \left (t \right )+25 \sin \left (t \right ) \sec \left (t \right )-10 \cos \left (t \right ) \sec \left (t \right )+5 \cos \left (t \right ) t \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: 0.021 (sec). Leaf size: 60
DSolve[{D[x1[t],t]==2*x1[t]-5*x2[t]+Sin[t],D[x2[t],t]==1*x1[t]-2*x2[t]+Tan[t]},{x1[0]==-1,x2[0]==0},{x1[t],x2[t]},t,IncludeSingularSolutions -> True]
\begin{align*}
\text {x1}(t)\to 5 \cos (t) \text {arctanh}(\sin (t))+\frac {1}{2} (t-12) \sin (t)-((t+1) \cos (t)) \\
\text {x2}(t)\to \text {arctanh}(\sin (t)) (\sin (t)+2 \cos (t))-\frac {5 \sin (t)}{2}-\frac {1}{2} t \cos (t)+\cos (t)-1 \\
\end{align*}