Internal problem ID [770]
Book: Elementary differential equations and boundary value problems, 10th ed., Boyce and
DiPrima
Section: Chapter 7.8, Repeated Eigenvalues. page 436
Problem number: 5.
ODE order: 1.
ODE degree: 1.
Solve \begin {align*} x_{1}^{\prime }\relax (t )&=x_{1} \relax (t )+x_{2} \relax (t )+x_{3} \relax (t )\\ x_{2}^{\prime }\relax (t )&=2 x_{1} \relax (t )+x_{2} \relax (t )-x_{3} \relax (t )\\ x_{3}^{\prime }\relax (t )&=-x_{2} \relax (t )+x_{3} \relax (t ) \end {align*}
✓ Solution by Maple
Time used: 0.079 (sec). Leaf size: 76
dsolve([diff(x__1(t),t)=1*x__1(t)+1*x__2(t)+1*x__3(t),diff(x__2(t),t)=2*x__1(t)+1*x__2(t)-1*x__3(t),diff(x__3(t),t)=0*x__1(t)-1*x__2(t)+1*x__3(t)],[x__1(t), x__2(t), x__3(t)], singsol=all)
\[ x_{1} \relax (t ) = -\frac {3 \,{\mathrm e}^{-t} c_{1}}{2}-c_{3} {\mathrm e}^{2 t} \] \[ x_{2} \relax (t ) = 2 \,{\mathrm e}^{-t} c_{1}-c_{2} {\mathrm e}^{2 t}-{\mathrm e}^{2 t} c_{3} t -c_{3} {\mathrm e}^{2 t} \] \[ x_{3} \relax (t ) = {\mathrm e}^{-t} c_{1}+c_{2} {\mathrm e}^{2 t}+{\mathrm e}^{2 t} c_{3} t \]
✓ Solution by Mathematica
Time used: 0.011 (sec). Leaf size: 151
DSolve[{x1'[t]==1*x1[t]+1*x2[t]+1*x3[t],x2'[t]==2*x1[t]+1*x2[t]-1*x3[t],x3'[t]==0*x1[t]-1*x2[t]+1*x3[t]},{x1[t],x2[t],x3[t]},t,IncludeSingularSolutions -> True]
\begin{align*} \text {x1}(t)\to \frac {1}{3} e^{-t} \left (c_1 \left (2 e^{3 t}+1\right )+(c_2+c_3) \left (e^{3 t}-1\right )\right ) \\ \text {x2}(t)\to \frac {1}{9} e^{-t} \left (e^{3 t} (c_1 (6 t+4)+c_2 (3 t+5)+c_3 (3 t-4))+4 (-c_1+c_2+c_3)\right ) \\ \text {x3}(t)\to \frac {1}{9} e^{-t} \left (2 (-c_1+c_2+c_3)-e^{3 t} (c_1 (6 t-2)+c_2 (3 t+2)+c_3 (3 t-7))\right ) \\ \end{align*}