I know nothing about diff. equations, is this one solvable?

[tamtam402]

I stumbled upon this diff. equation, which is a function of speed (v). Is there a technique I can use to solve it?

av(t) = b - c∫v(t)^2dt - d∫v(t)

Can I simply differentiate on both sides to get rid of the integral of the squared function, then use Laplace? The integrals go from 0 to an arbitrary time t.

Thanks!

 

[JJacquelin]

Can I simply differentiate on both sides to get rid of the integral of the squared function, then use Laplace?

Yes, you can. This will transform the integral equation to an ODE. But don't use Laplace transform because the ODE is not linear.
It's an ODE of the "separable" kind, directly leading to an integral easy to solve.
Don't forget to bring back your result into the original equation (not into the ODE) in order to compute the unknown constant.

 

[SteamKing]

It's not a differential equation, but an integral equation (yes, they have those, too._

 

[tamtam402]

Thanks for the reply guys! I figured out how to solve it by deriving on both sides, putting the v's and dv on one side and the dt on the other side then integrating again.

I still have a question. What made the first equation non-linear? I'm not sure why Laplace cannot be used here.

Thanks!

 

[JJacquelin]

What made the first equation non-linear? I'm not sure why Laplace cannot be used here.

In one of the terms, there is the square of the sought function. As a consequence the equation is not linear.
Try to use the Laplace transform of the square of a function. It involves Laplace convolution. If you know how to cope with it, then you can use Laplace transform.