Sketch the analogue of the trace-determinant plane in the bk-plane

Question

richyw · Answer

I'm confused as to what this means. It says "identify the relevant portion of the bk-plane where the corresponding system has similar phase portraits.

richyw · Answer

the differential equation is $x''+bx'+kx=0$ which I have written as

$$\left[\begin{matrix}x' \ y'\end{matrix}\right]=\left[\begin{matrix}0 & 1 \ -b & -k\end{matrix}\right]\left[\begin{matrix} x\y\end{matrix}\right]$$

richyw · Answer

and I have the characteristic equation$$\lambda^2+b\lambda+k=0$$$$\lambda=\frac{b}{2}\pm\sqrt{\frac{b^2}{4}-k}$$

experimentx · Answer

the eigen equation can be written as 
$$ λ2- \text{trace }λ+ \text{determinant }=0 $$

richyw · Answer

thanks, I think I hacked my way through it!

experimentx · Answer

congratulations ... there is a very good video lecture on differential equations about this on mit.ocw