Question

How do you integrate this differential equation then solve it? - Equation included

Assumption x=0 when v=0

Answer 1

Given: `v(dv)/dx = 9.8 - 0.004v^2, v(0) = 0`

Use the separation of variables method:

`v/(9.8 - 0.004v^2) dv = dx`

Integrate both sides:

`intv/(9.8 - 0.004v^2)dv = intdx`

Multiply by 1 in the form of `(-250)/-250:`

`int((-250)/-250)v/(9.8 - 0.004v^2) dv = intdx`

`-125int(2v)/(v^2-2450)dv = intdx`

Let `u = v^2` then `du = 2dv`:

`-125int1/(u-2450)du = intdx`

`ln(u-2450) = x/-125 + C`

Reverse the substitution:

`ln(v^2-2450) = x/-125 + C`

`v^2-2450 = e^(-0.008x + C)`

`v^2= 2450 + Ce^(-0.008x)`

Use the boundary condition:

`0^2 = 2450+ C`

`C = -2450`

`v^2= 2450 - 2450e^(-0.008x)`

`v^2= 2450(1 - e^(-0.008x))` Q.E.D.