Initial Value Problem for (DE)

[Vanessa Avila]

Homework Statement

dv/dt = 9.8 - (v/5) , v(0) = 0

(a) The time it must elapse for the objet to reach 98% of its limiting velocity
(b) How far does the object fall in the time found in part (a)?

Homework Equations

(dv/dt)/(9.8-(v/5))

The Attempt at a Solution

I'm a little overwhelmed by this class and I think the problem I have is I'm not catching on as to why the next answer is the way it is, so it would be nice if someone could explain to me why.
As I read on my textbook, I'm supposed to rewrite the form of the eqn first in which I attempted to do:

(dv/dt)/(9.8-(v/5)) = 1dt
but why do we have to rewrite the eqn in this form? I checked the solution for this, and it said it was right. I know afterwards you then integrate both sides, but I saw that they got
−5ln(9.8−(v/5))=t+C
I understand where the t+C comes from but not where the -5ln(9.8-(v/5)) comes from. Can someone explain to me how to get to that?

 

[Orodruin]

What is the integral of 1/x?

 

[Vanessa Avila]

What is the integral of 1/x?

ln(x)! Ah I missed that. Thanks!