Internal problem ID [1846]
Book: Differential equations and their applications, 4th ed., M. Braun
Section: Section 3.10, Systems of differential equations. Equal roots. Page 352
Problem number: 1.
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 )&=2 x_{1}\relax (t )-3 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: 0.065 (sec). Leaf size: 51
dsolve([diff(x__1(t),t)=0*x__1(t)-1*x__2(t)+1*x__3(t),diff(x__2(t),t)=2*x__1(t)-3*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)
\[ x_{1}\relax (t ) = {\mathrm e}^{-t} \left (c_{2} t +c_{1}\right ) \] \[ x_{2}\relax (t ) = {\mathrm e}^{-t} \left (c_{2} t +c_{1}\right )+c_{3} {\mathrm e}^{-2 t} \] \[ x_{3}\relax (t ) = c_{2} {\mathrm e}^{-t}+c_{3} {\mathrm e}^{-2 t} \]
✓ Solution by Mathematica
Time used: 0.007 (sec). Leaf size: 92
DSolve[{x1'[t]==0*x1[t]-1*x2[t]+1*x3[t],x2'[t]==2*x1[t]-3*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 e^{-t} (c_1 (t+1)+(c_3-c_2) t) \\ \text {x2}(t)\to e^{-2 t} \left (e^t (c_1 (t+1)+(c_3-c_2) t)-c_1+c_2\right ) \\ \text {x3}(t)\to e^{-2 t} \left ((c_1-c_2+c_3) e^t-c_1+c_2\right ) \\ \end{align*}