solve the differential equation dv/dt + (v^2)/90 = 9.8?

Answer 1

It's separable.

dv/dt = 9.8 - v²/90
dv/(9.8 - v²/90) = dt
dv/(882 - v²) = (1/90) dt
dv/[(√882 - v)(√882 + v)] = (1/90) dt

I would use partial fractions on the left side.

After integrating you'll get a couple of log terms on the left side. These can be combined, and the result can be solved for v as a function of t.

Answer 2

Write v = dv/ds, where s is the distance covered at time t.

then v dv/ds + (v^2)/90 = 9.8

dv/ds + v/90 = 9.8/v

This is a linear first order d.e.

It requires you to find the Integrating Factor exp(s/90), multiply throughout by that I.F. and proceed.

Answer 3

to make it easier for me to look at this im going to multiply through by 90

so we get 90dv/dt + v^2 = 882

as our answer is a constant our value of v must be a constant (which cancels the first order differentation of our function v)

so we get where c is a constant c^2 = 882

c = 29.7 (here c=v)