Donat el sistema lineal següent: \left\{ \begin{array}{l} \displaystyle \f...
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Donat el sistema lineal següent: \left\{ \begin{array}{l} \displaystyle \frac{dx}{dt} = x + y \\[8pt] \displaystyle \frac{dy}{dt} = -2x -y\\ \end{array} \right.
1) Digues quins són els valors propis
| λ = | ± | i |
|---|
(és a dir, λ1 = a + i b, λ2 = a - i b. Els vectors propis s'ordenen igual, el primer associat a λ1 (K1) i el segon a λ2 )
2) I els seus vectors propis associatsK1,2 =
| 1 ± i | ||
| ± i |
3) La solució general és
Per a x(t):
x(t) = exp(t) ( B1 [cos(t) + sin(t)] + B2 [cos(t) + sin(t)] )Per a y(t):
y(t) = exp(t) ( B1 [cos(t) + sin(t)] + B2 [ cos(t) + sin(t) ] )
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