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Math Help - help with riccati equation

  1. #1
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    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 donít understand how is simplifies. I donít know how to do intermediate steps, only the first and last. Can anyone helpÖ???
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  2. #2
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    Quote Originally Posted by mathprincess24 View Post
    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 donít understand how is simplifies. I donít know how to do intermediate steps, only the first and last. Can anyone helpÖ???
    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.
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  3. #3
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    I start by plugging in y = y1+ 1/v for all the yís 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 donít see any common things that would cancel. Am I on the right track and just not seeing it?
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  4. #4
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    can anyone tell me if i'm on the right track?
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