Constant forcing: free fall and the approach to terminal velocity under linear drag

Given: The parameters $m,g$ and the data of the two lines; the forced oscillator theorem applies in both since $k=0\ge 0$ and the forcing is constant, hence continuous.

For $m>0$, $c\ge 0$, $k\ge 0$ and continuous $f$, the unique solution of $m{x}^{\mathrm{\prime }\mathrm{\prime }}+c{x}^{\mathrm{\prime }}+kx=f$ with data $(x_0,v_0)$ at $t_0=0$ is $x=x_h+{\int }_0^{t}g(t-s)f(s)ds$, where $x_h$ is the homogeneous solution with the data and $g$ is the homogeneous kernel with $g(0)=0$, $g^{\mathrm{\prime }}(0)=1\mathrm{/}m$; for $k=c=0$ one has $g(u)=u\mathrm{/}m$, and for $k=0$, $c>0$ one has $g(u)=(1-e^{-cu\mathrm{/}m})\mathrm{/}c$ (The forced oscillator with $k\ge 0$: unique global solution and the convolution (Green's function) formula, The degenerate and repulsive oscillator: solutions for k=0 and the explicit growing solution for k<0). [given]

For $k=0$ the homogeneous solutions are $x_h(t)=A+B{e}^{-ct\mathrm{/}m}$ when $c>0$ and $x_h(t)=A+Bt$ when $c=0$, with $A=x_0+v_{0}m\mathrm{/}c$, $B=-v_{0}m\mathrm{/}c$ respectively $A=x_0$, $B=v_0$ (The degenerate and repulsive oscillator: solutions for k=0 and the explicit growing solution for k<0). [given]

$(e^{\mu t}{)}^{\mathrm{\prime }}=\mu e^{\mu t}$ for real $\mu$, and sums and scalar multiples are differentiated termwise. [thm-derivative-of-exponential, thm-algebra-of-derivatives]