Internal problem ID [787]
Book: Elementary differential equations and boundary value problems, 10th ed., Boyce and
DiPrima
Section: Chapter 7.9, Nonhomogeneous Linear Systems. page 447
Problem number: 10.
ODE order: 1.
ODE degree: 1.
Solve \begin {align*} x_{1}^{\prime }\relax (t )&=-3 x_{1} \relax (t )+\sqrt {2}\, x_{2} \relax (t )+{\mathrm e}^{-t}\\ x_{2}^{\prime }\relax (t )&=\sqrt {2}\, x_{1} \relax (t )-2 x_{2} \relax (t )-{\mathrm e}^{-t} \end {align*}
✓ Solution by Maple
Time used: 0.14 (sec). Leaf size: 91
dsolve([diff(x__1(t),t)=-3*x__1(t)+sqrt(2)*x__2(t)+exp(-t),diff(x__2(t),t)=sqrt(2)*x__1(t)-2*x__2(t)-exp(-t)],[x__1(t), x__2(t)], singsol=all)
\[ x_{1} \relax (t ) = \frac {t \,{\mathrm e}^{-t}}{3}+\frac {{\mathrm e}^{-t}}{3}-\sqrt {2}\, {\mathrm e}^{-4 t} c_{2}+\frac {\sqrt {2}\, {\mathrm e}^{-t} c_{1}}{2}-\frac {t \,{\mathrm e}^{-t} \sqrt {2}}{3}+\frac {{\mathrm e}^{-t} \sqrt {2}}{6} \] \[ x_{2} \relax (t ) = {\mathrm e}^{-4 t} c_{2}+{\mathrm e}^{-t} c_{1}+\frac {t \,{\mathrm e}^{-t} \sqrt {2}}{3}-\frac {2 t \,{\mathrm e}^{-t}}{3} \]
✓ Solution by Mathematica
Time used: 0.05 (sec). Leaf size: 128
DSolve[{x1'[t]==-3*x1[t]+Sqrt[2]*x2[t]+Exp[-t],x2'[t]==Sqrt[2]*x1[t]-2*x2[t]-Exp[-t]},{x1[t],x2[t]},t,IncludeSingularSolutions -> True]
\begin{align*} \text {x1}(t)\to \frac {1}{9} e^{-4 t} \left (e^{3 t} \left (-3 \left (\sqrt {2}-1\right ) t+\sqrt {2}+2+3 c_1+3 \sqrt {2} c_2\right )+6 c_1-3 \sqrt {2} c_2\right ) \\ \text {x2}(t)\to \frac {1}{9} e^{-4 t} \left (e^{3 t} \left (3 \left (\sqrt {2}-2\right ) t-\sqrt {2}-1+3 \sqrt {2} c_1+6 c_2\right )-3 \sqrt {2} c_1+3 c_2\right ) \\ \end{align*}