Internal problem ID [1835]
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: 12.
ODE order: 1.
ODE degree: 1.
Solve \begin {align*} x_{1}^{\prime }\relax (t )&=3 x_{1} \relax (t )+x_{2} \relax (t )-2 x_{3} \relax (t )\\ x_{2}^{\prime }\relax (t )&=-x_{1} \relax (t )+2 x_{2} \relax (t )+x_{3} \relax (t )\\ x_{3}^{\prime }\relax (t )&=4 x_{1} \relax (t )+x_{2} \relax (t )-3 x_{3} \relax (t ) \end {align*}
With initial conditions \[ [x_{1} \relax (0) = 1, x_{2} \relax (0) = 4, x_{3} \relax (0) = -7] \]
✓ Solution by Maple
Time used: 0.11 (sec). Leaf size: 58
dsolve([diff(x__1(t),t) = 3*x__1(t)+x__2(t)-2*x__3(t), diff(x__2(t),t) = -x__1(t)+2*x__2(t)+x__3(t), diff(x__3(t),t) = 4*x__1(t)+x__2(t)-3*x__3(t), x__1(0) = 1, x__2(0) = 4, x__3(0) = -7],[x__1(t), x__2(t), x__3(t)], singsol=all)
\[ x_{1} \relax (t ) = \frac {4 \,{\mathrm e}^{2 t}}{3}-\frac {28 \,{\mathrm e}^{-t}}{3}+9 \,{\mathrm e}^{t} \] \[ x_{2} \relax (t ) = \frac {4 \,{\mathrm e}^{2 t}}{3}+\frac {8 \,{\mathrm e}^{-t}}{3} \] \[ x_{3} \relax (t ) = \frac {4 \,{\mathrm e}^{2 t}}{3}-\frac {52 \,{\mathrm e}^{-t}}{3}+9 \,{\mathrm e}^{t} \]
✓ Solution by Mathematica
Time used: 0.01 (sec). Leaf size: 75
DSolve[{x1'[t]==3*x1[t]+1*x2[t]-2*x3[t],x2'[t]==-1*x1[t]+2*x2[t]+1*x3[t],x3'[t]==4*x1[t]+1*x2[t]-3*x3[t]},{x1[0]==1,x2[0]==4,x3[0]==-7},{x1[t],x2[t],x3[t]},t,IncludeSingularSolutions -> True]
\begin{align*} \text {x1}(t)\to -\frac {28 e^{-t}}{3}+9 e^t+\frac {4 e^{2 t}}{3} \\ \text {x2}(t)\to \frac {4}{3} e^{-t} \left (e^{3 t}+2\right ) \\ \text {x3}(t)\to -\frac {52 e^{-t}}{3}+9 e^t+\frac {4 e^{2 t}}{3} \\ \end{align*}