A confusion in Bernoulli equation solution

Question

I have a bernoulli equation with its proper and correct solution written step by step. The issue is that it contains a line that is transformed to the 2nd line which I was unable to understand how it is transformed. Here is the image of question and its solution, the problematic lines are highlighted in the image.

Click here to see the Solution

Please tell me how this transformation is done. thanks

Answer 1

The two lines are independent. One shows how the derivative is obtained and the next is the substitution.

Answer 2

The first line: You have y=v^-1/2, so

dy/dx = -1/2 V^-3/2 dv/dx

V^3/2 dy/dx =-1/2dv/dx

y^-3 dy/dx = -1/2 dv/dx

As for the second line:

y^-3 dy/dx = -1/2 dv/dx

dv/dx = -2y^-3 dy/dx =-2 (1+ y^-2) = -2-2y^-2 = -2-2v

So the equation is reduced to dv/dx + 2v = 2