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.
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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 $\displaystyle \frac{\partial f}{\partial t}= \frac{k}{v}$ which can still be integrated- except that the "constant of integration" may be a function of k.
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