# Trace Determinant Plane Differential Eqns

Consider the 2 parameter family of linear systems

$$\frac{DY(t)}{Dt} = \begin{pmatrix} a & 1 \\ b & 1 \end{pmatrix} Y(t)$$

In the ab plane, identify all regions where this system posseses a saddle, a sink, a spiral sink, and so on.

I was able to get the eigenvalues as $$\lambda = \frac{a+1}{2} \pm \frac{\sqrt{(a+1)^2 - 4(a-b)}}{2}$$

but need help in finding the sink and source.

I got the spiral sink as: if $a \lt -1$

spiral source if $a \gt -1$

and center if $a = -1$

Can someone check this?

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I've edited your post to be more readable, please go through it to know how it's done for next time :) – user864940 Aug 1 '11 at 7:15
Hint: What are the requirements of sink and source with respect to the eigenvalues? (Hint of hint: it has to do with the discriminant part of the quadratic formula.) – Willie Wong Aug 1 '11 at 13:59
Note the title of your question. The trace of the matrix is $a+1$ and the determinant is $a-b$. The conditions will probably be in terms of these two quantities and not just in terms of $a$. Also, $p^2-4q$ should play a role as well where $p$ is the trace and $q$ the determinant (this appears in your formula somewhere). – Matt Aug 2 '11 at 3:50

Summarizing the comments: the best way to begin is to look at determinant $a-b$ and trace $a+1$:

• $a-b<0$: saddle
• $a-b> 0$ and $a+1=0$: stable center
• $a-b> 0$ and $a+1<0$: stable node or spiral, depending on $(a+1)^2-4(a-b)$ being positive or negative
• $a-b> 0$ and $a+1>0$: unstable node or spiral, depending on $(a+1)^2-4(a-b)$ being positive or negative
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