a=dv/dt= kv
Separation variable: (1/v)dv= kdt.
Integral formula: lnv =kt+C brings initial condition t=0 v=v0, and the solution is C=lnv0.
Therefore: ln(v/v0)=kt, that is, v = v0e kt.
(2) Find the position-time relationship
v=dy/dt=v0e^kt
Separation variable: dy = (v0e kt) dt.
J integral: y = (v0/k) e kt+c brings in the initial condition of t=0 y=y0, and the solution is: C=y0-(v0/k).
Therefore, y = y0+(v0/k) (e kt- 1).