The Roberts' textbook Signals and Systems find the impulse response of a first-order differential equation using three different methods in Section 3.6 on pages 170-174.
A root locus plots the characteristic roots for different values of K and hence is a way to visualize what values of K lead to a stable system. The horizontal axis corresponds to the real part of the characteristic root, and the vertical axis corresponds to the imaginary part of the characteristic root.
The following Matlab code provides a simple way to generate a root locus plot. In Matlab, the polynomial s^2 + 3 s + 2 is represented as the row vector [1 3 2].
rootLocusPoints = ; for K = -10:1:10, pairs = roots( [1 3 K] ); %%% roots of polynomial s^2 + 3 s + K rootLocusPoints = [rootLocusPoints pairs(1) pairs(2)]; end scatter( real(rootLocusPoints), imag(rootLocusPoints) );
Matlab also has a root locus plotting called
which oddly enough implements the Evans root locus algorithm