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Thread: functions

  1. #1
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    functions

    Show that if f = f(x,y) and fyy = 0, then f(x,y) = g(x)y + h(x) for some functions g, h.
    (Start by stating the form taken by fy.)

    There is nothing in our notes to suggest what the significance of fyy=0 has, and even so I don't know where to start. Help please?
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  2. #2
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    Re: functions

    Quote Originally Posted by algebra123 View Post
    Show that if f = f(x,y) and fyy = 0, then f(x,y) = g(x)y + h(x) for some functions g, h.
    (Start by stating the form taken by fy.)

    If $\displaystyle f_{yy}=0$ then it is clear that $\displaystyle f_y=g(x)$ for some $\displaystyle g$.

    Hence $\displaystyle f(x,y)=g(x)y+h(x)$ for some $\displaystyle h$.
    Here $\displaystyle h(x)$ 'acts' as the constant $\displaystyle C$ in the indefinite anti-derivative.
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