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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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    yes, it is correct. Your only steady state is the origin, and the matrix is non-singular (due to the assumption on $a$), therefore, calculating the eigenvalues of your matrix will do.2012-05-14
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    @Artem i get the following for the eigenvalues though, $\lambda^{2}-3\lambda-2a=0$ ?2012-05-14
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    Nope, you missed something :)2012-05-14
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    @Artem can't see it :/2012-05-14
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    it should be $\lambda^2-3\lambda-2a-18=0$. Now you need to calculate the eigenvalues, and, depending on the found values, classify the fixed point at the origin.2012-05-14
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    @Artem ha yes i see i missed the -18 now, but i apply the quadratic formulae and get a=-9, which it can't be.2012-05-14
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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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