If I have a second order homogenous ODE
$$y''+(4+a)y'+(4+2a)y = 0 $$
for a constant $a$, the solutions come from the roots of the polynomial $x^2+(4+a)x+4+2a=0$, which are $x=-2$ and $x=-2-a$.
If $a\neq0$, the solutions are
$$y = c_1e^{-2x}+c_2e^{-(2+a)x}$$
This solution is not general though if I take the limit $a\rightarrow0$, as the root would become degenerate and the general solution would be
$$y_{a=0} = c_1e^{-2x}+c_2xe^{-2x}$$
Can I write the general solution for $y$ in a way in which I get the term proportional to $x$ if I take the limit $a\rightarrow0$?
When $a \neq 0$, $\{ e^{-2 x}, e^{-(2 - a)x} \}$ form a basis for the solutions of the linear homogeneous ODE.
When $a = 0$, $\{ e^{- 2 x}, x e^{- 2x} \}$ form a basis for the solutions of the linear homogeneous ODE in the degenerate case.