Internal problem ID [371]
Book: Differential equations and linear algebra, 4th ed., Edwards and Penney
Section: Section 7.6, Multiple Eigenvalue Solutions. Page 451
Problem number: problem 14.
ODE order: 1.
ODE degree: 1.
Solve \begin {align*} x_{1}^{\prime }\relax (t )&=x_{3}\relax (t )\\ x_{2}^{\prime }\relax (t )&=-5 x_{1}\relax (t )-x_{2}\relax (t )-5 x_{3}\relax (t )\\ x_{3}^{\prime }\relax (t )&=4 x_{1}\relax (t )+x_{2}\relax (t )-2 x_{3}\relax (t ) \end {align*}
✓ Solution by Maple
Time used: 0.021 (sec). Leaf size: 79
dsolve([diff(x__1(t),t)=0*x__1(t)+0*x__2(t)+1*x__3(t),diff(x__2(t),t)=-5*x__1(t)-1*x__2(t)-5*x__3(t),diff(x__3(t),t)=4*x__1(t)+1*x__2(t)-2*x__3(t)],[x__1(t), x__2(t), x__3(t)], singsol=all)
\[ x_{1}\relax (t ) = -{\mathrm e}^{-t} \left (c_{3} t^{2}+c_{2} t +2 c_{3} t +c_{1}+c_{2}+2 c_{3}\right ) \] \[ x_{2}\relax (t ) = {\mathrm e}^{-t} \left (5 c_{3} t^{2}+5 c_{2} t +10 c_{3} t +5 c_{1}+5 c_{2}+8 c_{3}\right ) \] \[ x_{3}\relax (t ) = {\mathrm e}^{-t} \left (c_{3} t^{2}+c_{2} t +c_{1}\right ) \]
✓ Solution by Mathematica
Time used: 0.003 (sec). Leaf size: 112
DSolve[{x1'[t]==0*x1[t]+0*x2[t]+1*x3[t],x2'[t]==-5*x1[t]-1*x2[t]-5*x3[t],x3'[t]==4*x1[t]+1*x2[t]-2*x3[t]},{x1[t],x2[t],x3[t]},t,IncludeSingularSolutions -> True]
\begin{align*} \text {x1}(t)\to \frac {1}{2} e^{-t} (c_1 (t (5 t+2)+2)+t (c_2 t+2 c_3)) \\ \text {x2}(t)\to \frac {1}{2} e^{-t} (2 c_2-5 t (c_1 (5 t+2)+c_2 t+2 c_3)) \\ \text {x3}(t)\to \frac {1}{2} e^{-t} (c_1 (8-5 t) t-c_2 (t-2) t-2 c_3 (t-1)) \\ \end{align*}