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

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Assumption x=0 when v=0

1 Answer
Jun 8, 2018

Given: v(dv)/dx = 9.8 - 0.004v^2, v(0) = 0vdvdx=9.80.004v2,v(0)=0

Use the separation of variables method:

v/(9.8 - 0.004v^2) dv = dxv9.80.004v2dv=dx

Integrate both sides:

intv/(9.8 - 0.004v^2)dv = intdxv9.80.004v2dv=dx

Multiply by 1 in the form of (-250)/-250:250250:

int((-250)/-250)v/(9.8 - 0.004v^2) dv = intdx(250250)v9.80.004v2dv=dx

-125int(2v)/(v^2-2450)dv = intdx1252vv22450dv=dx

Let u = v^2u=v2 then du = 2dvdu=2dv:

-125int1/(u-2450)du = intdx1251u2450du=dx

ln(u-2450) = x/-125 + Cln(u2450)=x125+C

Reverse the substitution:

ln(v^2-2450) = x/-125 + Cln(v22450)=x125+C

v^2-2450 = e^(-0.008x + C)v22450=e0.008x+C

v^2= 2450 + Ce^(-0.008x)v2=2450+Ce0.008x

Use the boundary condition:

0^2 = 2450+ C02=2450+C

C = -2450C=2450

v^2= 2450 - 2450e^(-0.008x)v2=24502450e0.008x

v^2= 2450(1 - e^(-0.008x))v2=2450(1e0.008x) Q.E.D.