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  • Jan 31, 2011, 04:40 PM
    kpg0001
    Linear Differential Equations
    Determine whether the given relation is an implicit solution to the given equation.

    sin(y)+xy-x^3=2-----relation

    y''=(6xy'+((y')^3)sin(y)-2(y')^2)/((3x^2)-y)-----equation

    The book says yes it is a solution but for some reason I am not getting it. You have to use implicit differentiation(duh).

    What I did, which should be the right steps, is get the second derivative and solve for it(go from y-y'-y'' and isolate y''). Maybe I'm missing something.

    This is what I got:

    y''=(2y'sin(y)-2y'+6x)/(cos(y)+x)
  • Jan 31, 2011, 10:14 PM
    jcaron2
    Using implicit differentiation, we should be able to find y' in terms of x and y:











    And differentiating a second time to find y'':





    Now we can go back to the equation for y' and rearrange a little:



    Plugging that back into the equation for y'', we get



    So I agree with your book. It definitely IS an implicit solution. Did that all make sense?

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