Question

sistema interessante...

Answer 1

En el primer sistema se deduce
$y_k'''-y_k=0\;,\; \text{with }k\in\{1,2,3\}$

$(D^3-1)y_k=0\\ \\ (D-1)(D^2+D+1)y_k=0\\ (r-1)(r^2+r+1)=0\iff r\in \left\{1,-\dfrac{1}{2}\pm \dfrac{\sqrt{3}}{2}i\right\}\\ \\ y_1=C_1e^{t}+C_2e^{-t/2}\cos \left(\dfrac{\sqrt{3}}{2}t\right)+C_3e^{-t/2}\sin \left(\dfrac{\sqrt{3}}{2}t\right)$

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derivamos la primera ecuación

$y_1'' = y_2'+y_3'=2y_1+y_2+y_3\\ \\ y_1'' =2y_1+y_1'\\ \\ y_1''-y_1'-2y_1=0$

$(D^2-D-2)y_1=0\\ \\ (D-2)(D+1)y_1=0\\ \\ (r-2)(r+1)=0\iff r\in\{-1,2\}\\ \\ \boxed{y_1=C_1e^{-t}+C_2e^{2t}}$