Internal problem ID [790]
Book: Elementary differential equations and boundary value problems, 10th ed., Boyce and
DiPrima
Section: Chapter 7.9, Nonhomogeneous Linear Systems. page 447
Problem number: 13.
ODE order: 1.
ODE degree: 1.
Solve \begin {align*} x_{1}^{\prime }\relax (t )&=-\frac {x_{1} \relax (t )}{2}-\frac {x_{2} \relax (t )}{8}+\frac {{\mathrm e}^{-\frac {t}{2}}}{2}\\ x_{2}^{\prime }\relax (t )&=2 x_{1} \relax (t )-\frac {x_{2} \relax (t )}{2} \end {align*}
✓ Solution by Maple
Time used: 0.062 (sec). Leaf size: 47
dsolve([diff(x__1(t),t)=-1/2*x__1(t)-1/8*x__2(t)+1/2*exp(-t/2),diff(x__2(t),t)=2*x__1(t)-1/2*x__2(t)+0],[x__1(t), x__2(t)], singsol=all)
\[ x_{1} \relax (t ) = \frac {{\mathrm e}^{-\frac {t}{2}} \left (c_{2} \cos \left (\frac {t}{2}\right )-c_{1} \sin \left (\frac {t}{2}\right )\right )}{4} \] \[ x_{2} \relax (t ) = {\mathrm e}^{-\frac {t}{2}} \left (\cos \left (\frac {t}{2}\right ) c_{1}+\sin \left (\frac {t}{2}\right ) c_{2}+4\right ) \]
✓ Solution by Mathematica
Time used: 0.018 (sec). Leaf size: 69
DSolve[{x1'[t]==-1/2*x1[t]-1/8*x2[t]+1/2*Exp[-t/2],x2'[t]==2*x1[t]-1/2*x2[t]+0},{x1[t],x2[t]},t,IncludeSingularSolutions -> True]
\begin{align*} \text {x1}(t)\to \frac {1}{4} e^{-t/2} \left (4 c_1 \cos \left (\frac {t}{2}\right )-c_2 \sin \left (\frac {t}{2}\right )\right ) \\ \text {x2}(t)\to e^{-t/2} \left (c_2 \cos \left (\frac {t}{2}\right )+4 c_1 \sin \left (\frac {t}{2}\right )+4\right ) \\ \end{align*}