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 }\left (t \right )&=2 x_{1} \left (t \right )-5 x_{2} \left (t \right )+\csc \left (t \right )\\ x_{2}^{\prime }\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: 0.39 (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)
\[ x_{1} \left (t \right ) = -5 \ln \left (\cos \left (t \right )\right ) \cos \left (t \right )+\cos \left (t \right ) \ln \left (\sin \left (t \right )\right )+2 \cos \left (t \right ) c_{1} +c_{2} \cos \left (t \right )-2 \cos \left (t \right ) t +2 \sin \left (t \right ) \ln \left (\sin \left (t \right )\right )-c_{1} \sin \left (t \right )+2 c_{2} \sin \left (t \right )-4 \sin \left (t \right ) t -2 \sin \left (t \right )-\sec \left (t \right )+\frac {\sin \left (t \right )^{2}}{\cos \left (t \right )} \] \[ x_{2} \left (t \right ) = -2 \ln \left (\cos \left (t \right )\right ) \cos \left (t \right )+\cos \left (t \right ) c_{1} -\ln \left (\cos \left (t \right )\right ) \sin \left (t \right )+\sin \left (t \right ) \ln \left (\sin \left (t \right )\right )+c_{2} \sin \left (t \right )-2 \sin \left (t \right ) t -\sin \left (t \right ) \]
✓ Solution by Mathematica
Time used: 0.018 (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*}