Internal problem ID [369]
Book: Differential equations and linear algebra, 4th ed., Edwards and Penney
Section: Section 7.6, Multiple Eigenvalue Solutions. Page 451
Problem number: problem 12.
ODE order: 1.
ODE degree: 1.
Solve \begin {align*} x_{1}^{\prime }\relax (t )&=-x_{1} \relax (t )+x_{3} \relax (t )\\ x_{2}^{\prime }\relax (t )&=-x_{2} \relax (t )+x_{3} \relax (t )\\ x_{3}^{\prime }\relax (t )&=x_{1} \relax (t )-x_{2} \relax (t )-x_{3} \relax (t ) \end {align*}
✓ Solution by Maple
Time used: 0.078 (sec). Leaf size: 62
dsolve([diff(x__1(t),t)=-1*x__1(t)+0*x__2(t)+1*x__3(t),diff(x__2(t),t)=0*x__1(t)-1*x__2(t)+1*x__3(t),diff(x__3(t),t)=1*x__1(t)-1*x__2(t)-1*x__3(t)],[x__1(t), x__2(t), x__3(t)], singsol=all)
\[ x_{1} \relax (t ) = \frac {{\mathrm e}^{-t} \left (t^{2} c_{3}+2 t c_{2}+2 c_{1}+2 c_{3}\right )}{2} \] \[ x_{2} \relax (t ) = \frac {\left (t^{2} c_{3}+2 t c_{2}+2 c_{1}\right ) {\mathrm e}^{-t}}{2} \] \[ x_{3} \relax (t ) = {\mathrm e}^{-t} \left (t c_{3}+c_{2}\right ) \]
✓ Solution by Mathematica
Time used: 0.003 (sec). Leaf size: 89
DSolve[{x1'[t]==-1*x1[t]+0*x2[t]+1*x3[t],x2'[t]==0*x1[t]-1*x2[t]+1*x3[t],x3'[t]==1*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}{2} e^{-t} \left (c_1 \left (t^2+2\right )+t (2 c_3-c_2 t)\right ) \\ \text {x2}(t)\to \frac {1}{2} e^{-t} \left ((c_1-c_2) t^2+2 c_3 t+2 c_2\right ) \\ \text {x3}(t)\to e^{-t} ((c_1-c_2) t+c_3) \\ \end{align*}