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 }\relax (t )&=2 x_{1}\relax (t )-5 x_{2}\relax (t )+\sin \relax (t )\\ x_{2}^{\prime }\relax (t )&=x_{1}\relax (t )-2 x_{2}\relax (t )+\tan \relax (t ) \end {align*}
With initial conditions \[ [x_{1}\relax (0) = 0, x_{2}\relax (0) = 0] \]
✓ Solution by Maple
Time used: 0.293 (sec). Leaf size: 282
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],[x__1(t), x__2(t)], singsol=all)
\[ x_{1}\relax (t ) = \left (-\frac {10 \sin \relax (t )}{-4 \sin \left (2 t \right )-5 \cos \relax (t )+\cos \left (3 t \right )}+\frac {10 \cos \left (4 t \right )-40 \sin \left (3 t \right )-40 \cos \left (2 t \right )-50}{-16 \sin \left (2 t \right )-20 \cos \relax (t )+4 \cos \left (3 t \right )}\right ) \ln \left (\frac {1+\sin \relax (t )}{\cos \relax (t )}\right )+\frac {\left (16 \sin \left (2 t \right )+20 \cos \relax (t )-4 \cos \left (3 t \right )\right ) \tan \relax (t )}{-16 \sin \left (2 t \right )-20 \cos \relax (t )+4 \cos \left (3 t \right )}+\frac {\left (-28+8 t \right ) \sin \relax (t )}{-16 \sin \left (2 t \right )-20 \cos \relax (t )+4 \cos \left (3 t \right )}+\frac {\left (32-4 t \right ) \cos \relax (t )-32 \cos \left (3 t \right )+4 \sin \left (3 t \right )+16 \cos \left (2 t \right )+48 \sin \left (2 t \right )-8 \sin \left (4 t \right )-16+\left (-2 \cos \left (4 t \right )+4 \cos \left (3 t \right )+8 \sin \left (3 t \right )+8 \cos \left (2 t \right )-6 \sin \left (2 t \right )+\sin \left (4 t \right )+10\right ) t}{-16 \sin \left (2 t \right )-20 \cos \relax (t )+4 \cos \left (3 t \right )} \] \[ x_{2}\relax (t ) = -\frac {3 \sin \relax (t )}{2}+\cos \relax (t )+\ln \left (\frac {1+\sin \relax (t )}{\cos \relax (t )}\right ) \sin \relax (t )+2 \ln \left (\frac {1+\sin \relax (t )}{\cos \relax (t )}\right ) \cos \relax (t )-\frac {\cos \relax (t ) t}{2}-1 \]
✓ Solution by Mathematica
Time used: 0.031 (sec). Leaf size: 142
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 \frac {1}{2} (t-8) \sin (t)-\cos (t) \left (t+5 \log \left (\cos \left (\frac {t}{2}\right )-\sin \left (\frac {t}{2}\right )\right )-5 \log \left (\sin \left (\frac {t}{2}\right )+\cos \left (\frac {t}{2}\right )\right )\right ) \\ \text {x2}(t)\to -\frac {1}{2} \cos (t) \left (t+4 \log \left (\cos \left (\frac {t}{2}\right )-\sin \left (\frac {t}{2}\right )\right )-4 \log \left (\sin \left (\frac {t}{2}\right )+\cos \left (\frac {t}{2}\right )\right )-2\right )+\sin (t) \left (-\log \left (\cos \left (\frac {t}{2}\right )-\sin \left (\frac {t}{2}\right )\right )+\log \left (\sin \left (\frac {t}{2}\right )+\cos \left (\frac {t}{2}\right )\right )-\frac {3}{2}\right )-1 \\ \end{align*}