Separation of variables: Context of decelerating charged particle

[CricK0es]

Homework Statement

I have attached an image of my problem. I realise that I need to use separation of variables to go from my LHS to my RHS; however, I’m unable to derive my RHS

Relevant Equations

*Attached in photo*

Attempted rewriting acceleration, a, in terms of dv/dt and then separating variables to integrate. This didn’t work...

So then I remembered that my gamma factor is also a function of v (!), but then I think I would be required to play around with integration by parts, which seems overly complicated.

Any help would be appreciated as I’m sure it’s straightforward but I’m really missing something here! Thank you

 

[pasmith]

The dependent variable is $v^2$. (That is the only context in which $\dots\,dt = \dots\,dv^2$ makes sense.) Therefore try multiplying $$-\frac{m\gamma^3}{\alpha} \frac{dv}{dt} = v$$ by $v$.

 

[CricK0es]

That’s perfect! I’ve never come across something where I’ve had to do it in that manner which is strange... But that makes a lot of sense. Thank you very much! I’ll post what I’ve done just to make sure I’ve got the right idea