Both pics are a bunch of hand-drawn equations and working out.
They are meaningless without the problem they belong to and the reasoning behind them, so I'll have to make some guesses in order to start helping you. In future, please try to describe the problem.
I can see a possible confusion in the line: $$k=\frac{1}{2k}\ln 3$$ ... have you used the same variable for two things here?
There is a red note: $$\frac{dy}{dx}+Py=a$$ ... if you didn't write that, then it may be a hint as to where you went wrong and you should look more closely at the DE you are trying to solve.
(I can't tell if that's supposed to be Py or Py)
$$ -v\frac{dv}{dk}=h+kv^2$$ ... is what you ended up with.
Assuming that is correct, it tidies up to:$$\frac{dv}{dk}+kv=-\frac{h}{v}$$
Which has form:$$\frac{dv}{dk}+p(k)v=q(v(k))$$... which you appear to have tried to solve via an integrating factor.
I'm guessing that this is what you want help with?
If so then: Compare with "Bernoulli's Equation".
If the writing in red is a hint, it appears to suggest that the initial DE is wrong... so the mistake is off the top of the page.
