Internal problem ID [1851]
Book: Differential equations and their applications, 4th ed., M. Braun
Section: Section 3.10, Systems of differential equations. Equal roots. Page 352
Problem number: 6.
ODE order: 1.
ODE degree: 1.
Solve \begin {align*} x_{1}^{\prime }\relax (t )&=-4 x_{1} \relax (t )-4 x_{2} \relax (t )\\ x_{2}^{\prime }\relax (t )&=10 x_{1} \relax (t )+9 x_{2} \relax (t )+x_{3} \relax (t )\\ x_{3}^{\prime }\relax (t )&=-4 x_{1} \relax (t )-3 x_{2} \relax (t )+x_{3} \relax (t ) \end {align*}
With initial conditions \[ [x_{1} \relax (0) = 2, x_{2} \relax (0) = 1, x_{3} \relax (0) = -1] \]
✓ Solution by Maple
Time used: 0.109 (sec). Leaf size: 57
dsolve([diff(x__1(t),t) = -4*x__1(t)-4*x__2(t), diff(x__2(t),t) = 10*x__1(t)+9*x__2(t)+x__3(t), diff(x__3(t),t) = -4*x__1(t)-3*x__2(t)+x__3(t), x__1(0) = 2, x__2(0) = 1, x__3(0) = -1],[x__1(t), x__2(t), x__3(t)], singsol=all)
\[ x_{1} \relax (t ) = {\mathrm e}^{2 t} \left (-4 t^{2}-16 t +2\right ) \] \[ x_{2} \relax (t ) = -{\mathrm e}^{2 t} \left (-6 t^{2}-26 t -1\right ) \] \[ x_{3} \relax (t ) = {\mathrm e}^{2 t} \left (-2 t^{2}-10 t -1\right ) \]
✓ Solution by Mathematica
Time used: 0.005 (sec). Leaf size: 57
DSolve[{x1'[t]==-4*x1[t]-4*x2[t]+0*x3[t],x2'[t]==10*x1[t]+9*x2[t]+1*x3[t],x3'[t]==-4*x1[t]-3*x2[t]+1*x3[t]},{x1[0]==2,x2[0]==1,x3[0]==-1},{x1[t],x2[t],x3[t]},t,IncludeSingularSolutions -> True]
\begin{align*} \text {x1}(t)\to -2 e^{2 t} (2 t (t+4)-1) \\ \text {x2}(t)\to e^{2 t} \left (6 t^2+26 t+1\right ) \\ \text {x3}(t)\to -e^{2 t} (2 t (t+5)+1) \\ \end{align*}