Help with basic problem of kinematics

[sleepwalker27]

Well, the problem says:
From some height a object with mass m is thrown . Determinate the law that describes how the velocity of fall v changes, if on the object, besides gravity, acts the force air resistance, which is proportional to velocity v (the proportionality coefficient is k), ie must be found v= f(t)

Then, by the second Newton's law, we have.
$$m\cdot\frac{dv}{dt}=mg-kv$$

AND THEN, HERE'S WHAT I DON'T UNDERSTAND. My textbook says:

Is easy to proof that every function:

$$v=Ce^{-\frac{k}{m}t}+\frac{mg}{k}$$

Satisfies the first equation.

Can somebody explain me in detail, why is that true? Thanks.

 

[sleepwalker27]

I think i asked de wrong question, the correct is: How the autor of the book came to that equation?

 

[Mr-R]

It is just a first order differential equation that can be solved by multiplying both sides of the equation by an integrating factor $U(t)$. Look it up. If you need more help let us know.
If you want to check (or proof) that it works, then just plug the solution $v$ into the differential equation.
The constant $C$ can be determine by providing the required initial conditions.

Edited: fixed a mistake regarding the method of solving the equation

 

[jedishrfu]

There's some history in the wiki article and while it doesn't directly answer your question may be helpful in finding the answer:

https://en.wikipedia.org/wiki/Differential_equation

I know when I first took this course it was like magic. I too wondered how these were discovered but alas it was never really discussed instead we were given recipes to identify the types of DEs and knowing that follow a few simple steps to gain the set of solutions.

Anyway from the article you can see how mathematicians like d'Alembert, Bernoulli, Lagrange and Euler took the codified knowledge of Calculus and extended to solve many common DE problems needed in physics.

 

[physics user1]

$$m\cdot\frac{dv}{dt}=mg-kv$$
Is a differential equation, if you solve it you get that function v (t)

 

[sleepwalker27]

Thank you all for your responses and your time, you helped me a lot.. I understood the problem