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
:
Monday, January 27, 2025 at 04:57:20 AM
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*}
✓ Solution by Maple
Time used: 0.308 (sec). Leaf size: 112
dsolve([diff(x__1(t),t)=2*x__1(t)-5*x__2(t)+csc(t),diff(x__2(t),t)=1*x__1(t)-2*x__2(t)+sec(t)],singsol=all)
\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 ) &= -2 \cos \left (t \right ) \ln \left (\cos \left (t \right )\right )+\frac {2 \cos \left (t \right ) c_1}{5}-\frac {c_2 \cos \left (t \right )}{5}+\ln \left (\sin \left (t \right )\right ) \sin \left (t \right )-\sin \left (t \right ) \ln \left (\cos \left (t \right )\right )+\frac {c_1 \sin \left (t \right )}{5}+\frac {2 \sin \left (t \right ) c_2}{5}-2 \sin \left (t \right ) t -\frac {\cos \left (t \right )^{2}}{5 \sin \left (t \right )}+\frac {2 \cos \left (t \right )}{5}+\frac {\csc \left (t \right )}{5} \\
\end{align*}
✓ Solution by Mathematica
Time used: 0.016 (sec). Leaf size: 79
DSolve[{D[ x1[t],t]==2*x1[t]-5*x2[t]+Csc[t],D[ x2[t],t]==1*x1[t]-2*x2[t]+Sec[t]},{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*}