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Consider the linear system

$\frac{dx}{dt}= -3x+2y, \frac{dy}{dt}= ax+6y, a \neq -9$

classify the fixed point at the origin?

Is the correct approach to investigate the steady states and how these points will change according to the point a, so consider the range from -$\infty$ to $\infty$.

Many thanks in advance.

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    It is hard to guess how you get $a=-9$. Your answer will depend on $a$, you shouldn't solve for $a$.2012-05-14

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It's not necessary to find eigenvalues explicitly: for $2\times 2$ matrices, the trace and determinant give enough information. Here $\mathrm{tr} = -3,\quad \det = -18-2a$ As $a$ moves from $-\infty$ to $\infty$, the matris moves down on the trace-determinant plane, from the region of stable spirals ($\mathrm{tr}^2-4\det<0$) to stable nodes ($\mathrm{tr}^2-4\det\ge 0$, $\det>0$) and finally to saddles ($\det<0$). We skip the inconvenient value $a=-9$, which would yield a non-isolated equilibrium ($\det=0$).