Thread: How to find the function v=f(t,k) for which dv/dt = k/v

Hi,

I'm looking for an expression where v is a function of t and k, and the derivative dv/dt = k/v.

Any hints in how to solve this? Thank you.

Originally Posted by fine1

Hi,

I'm looking for an expression where v is a function of t and k, and the derivative dv/dt = k/v.

Any hints in how to solve this? Thank you.

If k is merely a constant then the differential equation separates:
vdv = kdt

Simply integrate.

-Dan

Even if k is a function of t, you can still separate. Your solution will have an integral in it though...

Well, if f is a function of both k and t, then it should br which can still be integrated- except that the "constant of integration" may be a function of k.

Thanks to all for your help In my case k is not a function of t. The solution is simply v = squareroot(2kt).

Originally Posted by fine1

Thanks to all for your help In my case k is not a function of t. The solution is simply v = squareroot(2kt).

Actually that's only one possibitlity. You have left out the arbitrary constant...
v dv = k dt

(1/2)v^2 = kt + C

v = sqrt{2kt + 2C} = sqrt{2kt + A}
where A is arbitrary.

-Dan

Originally Posted by topsquark

Actually that's only one possibitlity. You have left out the arbitrary constant...
v dv = k dt

(1/2)v^2 = kt + C

v = sqrt{2kt + 2C} = sqrt{2kt + A}
where A is arbitrary.

-Dan

Quite so. Thanks