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

Question

Assumption x=0 when v=0

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Answer 1 · 2018-06-08T17:17:41

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.