Internal problem ID [1832]
Book: Differential equations and their applications, 4th ed., M. Braun
Section: Section 3.8, Systems of differential equations. The eigenva1ue-eigenvector method. Page
339
Problem number: 9.
ODE order: 1.
ODE degree: 1.
Solve \begin {align*} x_{1}^{\prime }\relax (t )&=3 x_{1} \relax (t )+x_{2} \relax (t )-x_{3} \relax (t )\\ x_{2}^{\prime }\relax (t )&=x_{1} \relax (t )+3 x_{2} \relax (t )-x_{3} \relax (t )\\ x_{3}^{\prime }\relax (t )&=3 x_{1} \relax (t )+3 x_{2} \relax (t )-x_{3} \relax (t ) \end {align*}
With initial conditions \[ [x_{1} \relax (0) = 1, x_{2} \relax (0) = -2, x_{3} \relax (0) = -1] \]
✓ Solution by Maple
Time used: 0.125 (sec). Leaf size: 27
dsolve([diff(x__1(t),t) = 3*x__1(t)+x__2(t)-x__3(t), diff(x__2(t),t) = x__1(t)+3*x__2(t)-x__3(t), diff(x__3(t),t) = 3*x__1(t)+3*x__2(t)-x__3(t), x__1(0) = 1, x__2(0) = -2, x__3(0) = -1],[x__1(t), x__2(t), x__3(t)], singsol=all)
\[ x_{1} \relax (t ) = {\mathrm e}^{2 t} \] \[ x_{2} \relax (t ) = -2 \,{\mathrm e}^{2 t} \] \[ x_{3} \relax (t ) = -{\mathrm e}^{2 t} \]
✓ Solution by Mathematica
Time used: 0.009 (sec). Leaf size: 30
DSolve[{x1'[t]==3*x1[t]+1*x2[t]-1*x3[t],x2'[t]==1*x1[t]+3*x2[t]-1*x3[t],x3'[t]==3*x1[t]+3*x2[t]-1*x3[t]},{x1[0]==1,x2[0]==-2,x3[0]==-1},{x1[t],x2[t],x3[t]},t,IncludeSingularSolutions -> True]
\begin{align*} \text {x1}(t)\to e^{2 t} \\ \text {x2}(t)\to -2 e^{2 t} \\ \text {x3}(t)\to -e^{2 t} \\ \end{align*}