Internal problem ID [789]
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.
ODE order: 1.
ODE degree: 1.
Solve \begin {align*} x_{1}^{\prime }\relax (t )&=-\frac {5 x_{2} \relax (t ) \sin \relax (t )-2 x_{1} \relax (t ) \sin \relax (t )-1}{\sin \relax (t )}\\ x_{2}^{\prime }\relax (t )&=-\frac {2 x_{2} \relax (t ) \cos \relax (t )-x_{1} \relax (t ) \cos \relax (t )-1}{\cos \relax (t )} \end {align*}
✓ Solution by Maple
Time used: 0.203 (sec). Leaf size: 113
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)],[x__1(t), x__2(t)], singsol=all)
\[ x_{1} \relax (t ) = -5 \cos \relax (t ) \ln \left (\cos \relax (t )\right )+\cos \relax (t ) \ln \left (\sin \relax (t )\right )+2 \sin \relax (t ) \ln \left (\sin \relax (t )\right )+2 \cos \relax (t ) c_{1}+c_{2} \cos \relax (t )-2 \cos \relax (t ) t -\sin \relax (t ) c_{1}+2 \sin \relax (t ) c_{2}-4 \sin \relax (t ) t -\sec \relax (t )-2 \sin \relax (t )+\frac {\sin ^{2}\relax (t )}{\cos \relax (t )} \] \[ x_{2} \relax (t ) = -2 \cos \relax (t ) \ln \left (\cos \relax (t )\right )-\sin \relax (t ) \ln \left (\cos \relax (t )\right )+\sin \relax (t ) \ln \left (\sin \relax (t )\right )+\cos \relax (t ) c_{1}+\sin \relax (t ) c_{2}-2 \sin \relax (t ) t -\sin \relax (t ) \]
✓ Solution by Mathematica
Time used: 0.022 (sec). Leaf size: 79
DSolve[{x1'[t]==2*x1[t]-5*x2[t]+Csc[t],x2'[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*}