14.25.5 problem 3
Internal
problem
ID
[2762]
Book
:
Differential
equations
and
their
applications,
4th
ed.,
M.
Braun
Section
:
Section
3.12,
Systems
of
differential
equations.
The
nonhomogeneous
equation.
variation
of
parameters.
Page
366
Problem
number
:
3
Date
solved
:
Tuesday, March 04, 2025 at 02:41:05 PM
CAS
classification
:
system_of_ODEs
\begin{align*} \frac {d}{d t}x_{1} \left (t \right )&=2 x_{1} \left (t \right )-5 x_{2} \left (t \right )+\sin \left (t \right )\\ \frac {d}{d t}x_{2} \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 ) = 0\\ x_{2} \left (0\right ) = 0 \end{align*}
✓ Maple. Time used: 0.068 (sec). Leaf size: 148
ode:=[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)];
ic:=x__1(0) = 0x__2(0) = 0;
dsolve([ode,ic]);
\begin{align*}
x_{1} \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 \\
x_{2} \left (t \right ) &= -\frac {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 ) \sec \left (t \right )-20 \cos \left (t \right ) \ln \left (\sec \left (t \right )+\tan \left (t \right )\right ) \sec \left (t \right )+15 \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 \tan \left (t \right )^{2} \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 )+15 \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 \cos \left (t \right )}{10 \left (\sec \left (t \right )+\tan \left (t \right )\right )} \\
\end{align*}
✓ Mathematica. Time used: 0.021 (sec). Leaf size: 58
ode={D[ x1[t],t]==2*x1[t]-5*x2[t]+Sin[t],D[ x2[t],t]==1*x1[t]-2*x2[t]+Tan[t]};
ic={x1[0]==0,x2[0]==0};
DSolve[{ode,ic},{x1[t],x2[t]},t,IncludeSingularSolutions->True]
\begin{align*}
\text {x1}(t)\to 5 \cos (t) \text {arctanh}(\sin (t))+\frac {1}{2} (t-8) \sin (t)-t \cos (t) \\
\text {x2}(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*}
✓ Sympy. Time used: 0.213 (sec). Leaf size: 162
from sympy import *
t = symbols("t")
x__1 = Function("x__1")
x__2 = Function("x__2")
ode=[Eq(-2*x__1(t) + 5*x__2(t) - sin(t) + Derivative(x__1(t), t),0),Eq(-x__1(t) + 2*x__2(t) - tan(t) + Derivative(x__2(t), t),0)]
ics = {}
dsolve(ode,func=[x__1(t),x__2(t)],ics=ics)
\[
\left [ x^{1}{\left (t \right )} = \frac {t \sin {\left (t \right )}}{2} - t \cos {\left (t \right )} - \left (C_{1} - 2 C_{2}\right ) \cos {\left (t \right )} - \left (2 C_{1} + C_{2}\right ) \sin {\left (t \right )} - \frac {5 \log {\left (\sin {\left (t \right )} - 1 \right )} \cos {\left (t \right )}}{2} + \frac {5 \log {\left (\sin {\left (t \right )} + 1 \right )} \cos {\left (t \right )}}{2} + \sin ^{3}{\left (t \right )} + \sin {\left (t \right )} \cos ^{2}{\left (t \right )}, \ x^{2}{\left (t \right )} = - C_{1} \sin {\left (t \right )} + C_{2} \cos {\left (t \right )} - \frac {t \cos {\left (t \right )}}{2} - \frac {\log {\left (\sin {\left (t \right )} - 1 \right )} \sin {\left (t \right )}}{2} - \log {\left (\sin {\left (t \right )} - 1 \right )} \cos {\left (t \right )} + \frac {\log {\left (\sin {\left (t \right )} + 1 \right )} \sin {\left (t \right )}}{2} + \log {\left (\sin {\left (t \right )} + 1 \right )} \cos {\left (t \right )} + \frac {\sin ^{3}{\left (t \right )}}{2} - \sin ^{2}{\left (t \right )} + \frac {\sin {\left (t \right )} \cos ^{2}{\left (t \right )}}{2} - \cos ^{2}{\left (t \right )}\right ]
\]