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

1. ## 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.

2. 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

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

4. Well, if f is a function of both k and t, then it should br $\frac{\partial f}{\partial t}= \frac{k}{v}$ which can still be integrated- except that the "constant of integration" may be a function of k.

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

6. 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

7. 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