Thread: help with riccati equation

dy/dx = A(x)y2 + B(x)y + C(x) is called a riccati equation. Suppose that one particular solution y1(x) of this equation is known. Show that the substitution
Y= y1 + 1/v transforms the riccati equation into the linear equation:
Dv/dx + (B(x) +2A(x)y1)v = -A(x)

I understand what you need to do. I know that I need to plug in the substitution for y, but I don’t understand how is simplifies. I don’t know how to do intermediate steps, only the first and last. Can anyone help…???

Originally Posted by mathprincess24

dy/dx = A(x)y2 + B(x)y + C(x) is called a riccati equation. Suppose that one particular solution y1(x) of this equation is known. Show that the substitution
Y= y1 + 1/v transforms the riccati equation into the linear equation:
Dv/dx + (B(x) +2A(x)y1)v = -A(x)

I understand what you need to do. I know that I need to plug in the substitution for y, but I don’t understand how is simplifies. I don’t know how to do intermediate steps, only the first and last. Can anyone help…???

Show what you have done so far, please. perhaps you are just forgetting that y1 is a solution to the original equation so that many things will cancel.

I start by plugging in y = y1+ 1/v for all the y’s in the original equation. Then I get:
Dv/dx = A(x) ( y1+ 1/v)2 + B(x)( y1+ 1/v) + C(x)
I tried expanding the squared term and distributing, but I don’t see any common things that would cancel. Am I on the right track and just not seeing it?

can anyone tell me if i'm on the right track?