Internal problem ID [1858]
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.
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 )+\sin \left (t \right )\\ x_{2}^{\prime }\left (t \right )&=x_{1} \left (t \right )-2 x_{2} \left (t \right )+\tan \left (t \right ) \end {align*}
With initial conditions \[ [x_{1} \left (0\right ) = 0, x_{2} \left (0\right ) = 0] \]
✓ Solution by Maple
Time used: 0.187 (sec). Leaf size: 149
dsolve([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), x__1(0) = 0, x__2(0) = 0], singsol=all)
\begin{align*} x_{1} \left (t \right ) &= -4 \sin \left (t \right )+5 \cos \left (t \right ) \ln \left (\sec \left (t \right )+\tan \left (t \right )\right )-\cos \left (t \right ) t +\frac {\sin \left (t \right ) t}{2} \\ x_{2} \left (t \right ) &= \frac {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 \sec \left (t \right ) \cos \left (t \right ) \tan \left (t \right )+10 \cos \left (t \right ) \sec \left (t \right )-5 \cos \left (t \right ) t \sec \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 )^{2}+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 \sec \left (t \right )+10 \tan \left (t \right )} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.069 (sec). Leaf size: 58
DSolve[{x1'[t]==2*x1[t]-5*x2[t]+Sin[t],x2'[t]==1*x1[t]-2*x2[t]+Tan[t]},{x1[0]==0,x2[0]==0},{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*}