Internal problem ID [771]
Book: Elementary differential equations and boundary value problems, 10th ed., Boyce and
DiPrima
Section: Chapter 7.8, Repeated Eigenvalues. page 436
Problem number: 6.
ODE order: 1.
ODE degree: 1.
Solve \begin {align*} x_{1}^{\prime }\relax (t )&=x_{2} \relax (t )+x_{3} \relax (t )\\ x_{2}^{\prime }\relax (t )&=x_{1} \relax (t )+x_{3} \relax (t )\\ x_{3}^{\prime }\relax (t )&=x_{1} \relax (t )+x_{2} \relax (t ) \end {align*}
✓ Solution by Maple
Time used: 0.078 (sec). Leaf size: 64
dsolve([diff(x__1(t),t)=0*x__1(t)+1*x__2(t)+1*x__3(t),diff(x__2(t),t)=1*x__1(t)+0*x__2(t)+1*x__3(t),diff(x__3(t),t)=1*x__1(t)+1*x__2(t)+0*x__3(t)],[x__1(t), x__2(t), x__3(t)], singsol=all)
\[ x_{1} \relax (t ) = c_{2} {\mathrm e}^{2 t}-2 c_{3} {\mathrm e}^{-t}-{\mathrm e}^{-t} c_{1} \] \[ x_{2} \relax (t ) = c_{2} {\mathrm e}^{2 t}+c_{3} {\mathrm e}^{-t}+{\mathrm e}^{-t} c_{1} \] \[ x_{3} \relax (t ) = c_{2} {\mathrm e}^{2 t}+c_{3} {\mathrm e}^{-t} \]
✓ Solution by Mathematica
Time used: 0.006 (sec). Leaf size: 114
DSolve[{x1'[t]==0*x1[t]+1*x2[t]+1*x3[t],x2'[t]==1*x1[t]+0*x2[t]+1*x3[t],x3'[t]==1*x1[t]+1*x2[t]+0*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 (e^{3 t}+2\right )+(c_2+c_3) \left (e^{3 t}-1\right )\right ) \\ \text {x2}(t)\to \frac {1}{3} e^{-t} \left ((c_1+c_2+c_3) e^{3 t}-c_1+2 c_2-c_3\right ) \\ \text {x3}(t)\to \frac {1}{3} e^{-t} \left ((c_1+c_2+c_3) e^{3 t}-c_1-c_2+2 c_3\right ) \\ \end{align*}