Linearize the equation $$x'' = -\alpha x-\rho x'+c \sin(t)$$
It is very easy when $c=0$ giving you a $$ x' = y $$$$ y' = -\alpha x -\rho y $$ giving you a very nice phase portrait.
However, if $c$ is non-zero, the linearization should be like $$ x' = y $$$$ y' = -\alpha x -\rho y +c\sin(t) $$ but this gives a very ugly phase portrait (the lines keep intersecting with themselves)
Is this still accurate or should I further use a Jacobian to linearize the matrix? (If so, can somebody provide a small hint on how to approach the result, take any initial state for initial condition)