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For instance

dv/dt = a

dv = a*dt

v= a*t + c (c is a constant of integration)

then if I was given an initial condition such as v(0)=Vo I would substitute into my general solution

and get

Vo=a*0+c=c

V=a*t+Vo

Now that I started learning physics I keep coming across this funny way of solving differential equations:

v

[inte] dv=

Vo

t

[inte] a*dt

0

Why does this second method work. I realize that it is probably somehow equivalent to the first I just don’t see it. I would really like to know how and why this second method works.