10.18.12 problem 12
Internal
problem
ID
[1439]
Book
:
Elementary
differential
equations
and
boundary
value
problems,
10th
ed.,
Boyce
and
DiPrima
Section
:
Chapter
7.9,
Nonhomogeneous
Linear
Systems.
page
447
Problem
number
:
12
Date
solved
:
Saturday, March 29, 2025 at 10:55:15 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 )+\csc \left (t \right )\\ \frac {d}{d t}x_{2} \left (t \right )&=x_{1} \left (t \right )-2 x_{2} \left (t \right )+\sec \left (t \right ) \end{align*}
✓ Maple. Time used: 0.511 (sec). Leaf size: 143
ode:=[diff(x__1(t),t) = 2*x__1(t)-5*x__2(t)+csc(t), diff(x__2(t),t) = x__1(t)-2*x__2(t)+sec(t)];
dsolve(ode);
\begin{align*}
x_{1} \left (t \right ) &= \ln \left (\sin \left (t \right )\right ) \cos \left (t \right )-5 \cos \left (t \right ) \ln \left (\cos \left (t \right )\right )+\cos \left (t \right ) c_1 -2 \cos \left (t \right ) t +2 \ln \left (\sin \left (t \right )\right ) \sin \left (t \right )+\sin \left (t \right ) c_2 -4 \sin \left (t \right ) t +\cos \left (t \right ) \\
x_{2} \left (t \right ) &= -\frac {\left (10 \ln \left (\cos \left (t \right )\right ) \sin \left (2 t \right )-2 c_1 \sin \left (2 t \right )+c_2 \sin \left (2 t \right )+5 \ln \left (\sin \left (t \right )\right ) \cos \left (2 t \right )-5 \ln \left (\cos \left (t \right )\right ) \cos \left (2 t \right )+c_1 \cos \left (2 t \right )+2 c_2 \cos \left (2 t \right )-10 t \cos \left (2 t \right )-2 \sin \left (2 t \right )+\cos \left (2 t \right )-5 \ln \left (\sin \left (t \right )\right )+5 \ln \left (\cos \left (t \right )\right )-c_1 -2 c_2 +10 t -1\right ) \csc \left (t \right )}{10} \\
\end{align*}
✓ Mathematica. Time used: 0.016 (sec). Leaf size: 79
ode={D[ x1[t],t]==2*x1[t]-5*x2[t]+Csc[t],D[ x2[t],t]==1*x1[t]-2*x2[t]+Sec[t]};
ic={};
DSolve[{ode,ic},{x1[t],x2[t]},t,IncludeSingularSolutions->True]
\begin{align*}
\text {x1}(t)\to \sin (t) (-4 t+2 \log (\sin (t))+2 c_1-5 c_2)+\cos (t) (-2 t+\log (\sin (t))-5 \log (\cos (t))+c_1) \\
\text {x2}(t)\to \cos (t) (-2 \log (\cos (t))+c_2)+\sin (t) (-2 t+\log (\sin (t))-\log (\cos (t))+c_1-2 c_2) \\
\end{align*}
✓ Sympy. Time used: 0.163 (sec). Leaf size: 107
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) + Derivative(x__1(t), t) - 1/sin(t),0),Eq(-x__1(t) + 2*x__2(t) + Derivative(x__2(t), t) - 1/cos(t),0)]
ics = {}
dsolve(ode,func=[x__1(t),x__2(t)],ics=ics)
\[
\left [ x^{1}{\left (t \right )} = - 4 t \sin {\left (t \right )} - 2 t \cos {\left (t \right )} - \left (C_{1} - 2 C_{2}\right ) \cos {\left (t \right )} - \left (2 C_{1} + C_{2}\right ) \sin {\left (t \right )} + 2 \log {\left (\sin {\left (t \right )} \right )} \sin {\left (t \right )} + \log {\left (\sin {\left (t \right )} \right )} \cos {\left (t \right )} - 5 \log {\left (\cos {\left (t \right )} \right )} \cos {\left (t \right )}, \ x^{2}{\left (t \right )} = - C_{1} \sin {\left (t \right )} + C_{2} \cos {\left (t \right )} - 2 t \sin {\left (t \right )} + \log {\left (\sin {\left (t \right )} \right )} \sin {\left (t \right )} - \log {\left (\cos {\left (t \right )} \right )} \sin {\left (t \right )} - 2 \log {\left (\cos {\left (t \right )} \right )} \cos {\left (t \right )}\right ]
\]