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Math Help - Small question about Riccati

  1. #1
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    Small question about Riccati

    The form of Riccati is y' = P(x)y^{2} + Q(x)y + R(x)
    the book says that y' = \frac{1}{x^{2}}y^{2} - \frac{1}{x}y + 1 is a Riccati equation.
    So how does the constant 1 qualify as a R(x) term?
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    Re: Small question about Riccati

    Quote Originally Posted by catenary View Post
    The form of Riccati is y' = P(x)y^{2} + Q(x)y + R(x)
    the book says that y' = \frac{1}{x^{2}}y^{2} - \frac{1}{x}y + 1 is a Riccati equation.
    So how does the constant 1 qualify as a R(x) term?
    1 is a perfectly legitimate function of x.

    $R(x)=1$
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    Re: Small question about Riccati

    is it a legitimate function of x because you can write it as 1 = 1(x)^{0} ?
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    Re: Small question about Riccati

    Quote Originally Posted by catenary View Post
    is it a legitimate function of x because you can write it as 1 = 1(x)^{0} ?
    sure if you like but even if you couldn't it's still a function of x. It just happens to be a constant function.

    Does it map every value in it's domain to a unique number? Yes. It maps every value in it's domain to 1. That's all you need to be a function.
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