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.