Internal problem ID [1842]
Book: Differential equations and their applications, 4th ed., M. Braun
Section: Section 3.9, Systems of differential equations. Complex roots. Page 344
Problem number: 7.
ODE order: 1.
ODE degree: 1.
Solve \begin {align*} x_{1}^{\prime }\left (t \right )&=-3 x_{1} \left (t \right )+2 x_{3} \left (t \right )\\ x_{2}^{\prime }\left (t \right )&=x_{1} \left (t \right )-x_{2} \left (t \right )\\ x_{3}^{\prime }\left (t \right )&=-2 x_{1} \left (t \right )-x_{2} \left (t \right ) \end {align*}
With initial conditions \[ [x_{1} \left (0\right ) = 0, x_{2} \left (0\right ) = -1, x_{3} \left (0\right ) = -2] \]
✓ Solution by Maple
Time used: 0.031 (sec). Leaf size: 95
dsolve([diff(x__1(t),t) = -3*x__1(t)+2*x__3(t), diff(x__2(t),t) = x__1(t)-x__2(t), diff(x__3(t),t) = -2*x__1(t)-x__2(t), x__1(0) = 0, x__2(0) = -1, x__3(0) = -2], singsol=all)
\begin{align*} x_{1} \left (t \right ) &= 2 \,{\mathrm e}^{-2 t}-\sqrt {2}\, {\mathrm e}^{-t} \sin \left (\sqrt {2}\, t \right )-2 \,{\mathrm e}^{-t} \cos \left (\sqrt {2}\, t \right ) \\ x_{2} \left (t \right ) &= -2 \,{\mathrm e}^{-2 t}+{\mathrm e}^{-t} \cos \left (\sqrt {2}\, t \right )-\sqrt {2}\, {\mathrm e}^{-t} \sin \left (\sqrt {2}\, t \right ) \\ x_{3} \left (t \right ) &= {\mathrm e}^{-2 t}-3 \,{\mathrm e}^{-t} \cos \left (\sqrt {2}\, t \right ) \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.043 (sec). Leaf size: 109
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[0]==0,x2[0]==-1,x3[0]==-2},{x1[t],x2[t],x3[t]},t,IncludeSingularSolutions -> True]
\begin{align*} \text {x1}(t)\to -e^{-2 t} \left (\sqrt {2} e^t \sin \left (\sqrt {2} t\right )+2 e^t \cos \left (\sqrt {2} t\right )-2\right ) \\ \text {x2}(t)\to e^{-2 t} \left (-\sqrt {2} e^t \sin \left (\sqrt {2} t\right )+e^t \cos \left (\sqrt {2} t\right )-2\right ) \\ \text {x3}(t)\to e^{-2 t} \left (1-3 e^t \cos \left (\sqrt {2} t\right )\right ) \\ \end{align*}