I am studying Hirsch, Smale, Devaney; Differential equations, dynamical systems, and an introduction to chaos (2nd edition) and have a couple of questions.
First, in Section 3.3 about repeated eigenvalues, we study
$$\begin{pmatrix}x\\y\end{pmatrix}'=\begin{pmatrix}\lambda&1\\0&\lambda\end{pmatrix}\begin{pmatrix}x\\y\end{pmatrix}.$$
The solution is basically solve the second line, getting $y(t)=\beta e^{\lambda t}$ (*). So, $x'(t)=\lambda x(t)+\beta e^{\lambda t}$ and we can use the method of undetermined coefficients, getting $x'(t)=\alpha e^{\lambda t}+\mu t e^{\lambda t}.$
I was wondering:
And if neither of the two lines is independent of the other? I mean, and if I cannot get (*)?
The second question is about the trace-determinant plane. The book says:
A one-parameter family of linear systems corresponds to a curve in the TD–plane. When this curve crosses the T-axis, the positive D-axis, or the parabola $T^2 − 4D = 0$, the phase portrait of the linear system undergoes a bifurcation: A major change occurs in the geometry of the phase portrait.
However, I was wondering something like this:
Where is my error?
Thank you in advance!