Math Help - Show that the Riccati equation boils down to linear form in v

I've been trying this problem but in both ways that i try it i get stuck. i would like to get any hints onto how to start it.

An equation: dy/dx = P(x)y^2 + Q(x)y + R(x) is called the riccati form...

If one solution, say u(x), of this is known, show that the substitution y=u+(1/v) reduces the riccati equation into a linear equation in v.

Thnak u very much!

Well, what have you tried so far?

First I start with y=u+(1/v) and from this I get: dy/dx = u' - (1/v^2)(dv/dx)

Then I put these last expression into the given riccati form.

From here I tried 2 things:
1) In the riccati form I substituded y by u(x)... this led me to call P(x)u^2 + Q(x)u + R(x) = -F(x) ... and u' = G(x)
From here I have dv/dx + v^2G(x) = v^2F(x) ... The problem is that this is not linear, and not even bernuolli seems to be available.

2) I substitute y=u+(1/v) in the riccati form, but it gets super ugly and it doesnt seem to go anywhere.

Thanks!

Hmm. You have with a known particular solution. Your goal is to show that the substitution renders the equation linear in Your derivative is correct. Now plug everything into the DE:



Rearranging a bit yields



Can you finish from here?