Internal problem ID [758]
Book: Elementary differential equations and boundary value problems, 10th ed., Boyce and
DiPrima
Section: Chapter 7.6, Complex Eigenvalues. page 417
Problem number: 8.
ODE order: 1.
ODE degree: 1.
Solve \begin {align*} x_{1}^{\prime }\relax (t )&=-3 x_{1}\relax (t )+2 x_{3}\relax (t )\\ x_{2}^{\prime }\relax (t )&=x_{1}\relax (t )-x_{2}\relax (t )\\ x_{3}^{\prime }\relax (t )&=-2 x_{1}\relax (t )-x_{2}\relax (t ) \end {align*}
✓ Solution by Maple
Time used: 0.052 (sec). Leaf size: 174
dsolve([diff(x__1(t),t)=-3*x__1(t)+0*x__2(t)+2*x__3(t),diff(x__2(t),t)=1*x__1(t)-1*x__2(t)-0*x__3(t),diff(x__3(t),t)=-2*x__1(t)-1*x__2(t)+0*x__3(t)],[x__1(t), x__2(t), x__3(t)], singsol=all)
\[ x_{1}\relax (t ) = 2 c_{1} {\mathrm e}^{-2 t}+\frac {2 c_{2} {\mathrm e}^{-t} \sin \left (t \sqrt {2}\right )}{3}-\frac {c_{2} {\mathrm e}^{-t} \sqrt {2}\, \cos \left (t \sqrt {2}\right )}{3}+\frac {2 c_{3} {\mathrm e}^{-t} \cos \left (t \sqrt {2}\right )}{3}+\frac {c_{3} {\mathrm e}^{-t} \sqrt {2}\, \sin \left (t \sqrt {2}\right )}{3} \] \[ x_{2}\relax (t ) = -2 c_{1} {\mathrm e}^{-2 t}-\frac {c_{2} {\mathrm e}^{-t} \sin \left (t \sqrt {2}\right )}{3}-\frac {c_{2} {\mathrm e}^{-t} \sqrt {2}\, \cos \left (t \sqrt {2}\right )}{3}-\frac {c_{3} {\mathrm e}^{-t} \cos \left (t \sqrt {2}\right )}{3}+\frac {c_{3} {\mathrm e}^{-t} \sqrt {2}\, \sin \left (t \sqrt {2}\right )}{3} \] \[ x_{3}\relax (t ) = c_{1} {\mathrm e}^{-2 t}+c_{2} {\mathrm e}^{-t} \sin \left (t \sqrt {2}\right )+c_{3} {\mathrm e}^{-t} \cos \left (t \sqrt {2}\right ) \]
✓ Solution by Mathematica
Time used: 0.04 (sec). Leaf size: 231
DSolve[{x1'[t]==-3*x1[t]+0*x2[t]+2*x3[t],x2'[t]==1*x1[t]-1*x2[t]-0*x3[t],x3'[t]==-2*x1[t]-1*x2[t]+0*x3[t]},{x1[t],x2[t],x3[t]},t,IncludeSingularSolutions -> True]
\begin{align*} \text {x1}(t)\to \frac {1}{3} e^{-2 t} \left (e^t \left ((c_1+2 (c_2+c_3)) \cos \left (\sqrt {2} t\right )-\sqrt {2} (2 c_1+c_2-2 c_3) \sin \left (\sqrt {2} t\right )\right )+2 (c_1-c_2-c_3)\right ) \\ \text {x2}(t)\to \frac {1}{6} e^{-2 t} \left (e^t \left (2 (2 c_1+c_2-2 c_3) \cos \left (\sqrt {2} t\right )+\sqrt {2} (c_1+2 (c_2+c_3)) \sin \left (\sqrt {2} t\right )\right )+4 (-c_1+c_2+c_3)\right ) \\ \text {x3}(t)\to \frac {1}{6} e^{-2 t} \left (e^t \left (2 (-c_1+c_2+4 c_3) \cos \left (\sqrt {2} t\right )+\sqrt {2} (-5 c_1-4 c_2+2 c_3) \sin \left (\sqrt {2} t\right )\right )+2 (c_1-c_2-c_3)\right ) \\ \end{align*}