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Math Help - Solving Bernoulli equations

  1. #1
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    Unhappy Solving Bernoulli equations

    I am in a class on differential equations and do not understand how to solve Bernoulli equations. Here is the problem I have been working on:

    y' = ry - k(y^2) , r>0 , k>0

    So far I have divided both sides by y^2, and rearranged the equation so that it looks like this:

    (y')/(y^2) = (ry)/(y^2) - k(y^2)/(y^2)

    (y')/(y^2) = (ry)/(y^2) - k

    y'(y^-2) = r(y^-1) - k

    y'(y^-2) - r(y^-1) = -k

    Could some please help? I have a test tomorrow and I am completely stuck.
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  2. #2
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    Quote Originally Posted by CatalinaLou View Post
    I am in a class on differential equations and do not understand how to solve Bernoulli equations. Here is the problem I have been working on:

    y' = ry - k(y^2) , r>0 , k>0

    So far I have divided both sides by y^2, and rearranged the equation so that it looks like this:

    (y')/(y^2) = (ry)/(y^2) - k(y^2)/(y^2)

    (y')/(y^2) = (ry)/(y^2) - k

    y'(y^-2) = r(y^-1) - k

    y'(y^-2) - r(y^-1) = -k

    Could some please help? I have a test tomorrow and I am completely stuck.
    So
    y' = ry - k y^2

    then

    \frac{y' }{y^2}= \frac{r}{y} - k

    If u = \frac{1}{y} then u' = - \frac{y'}{y^2}

    and your equation becomes

    - u' = r u - k

    or

     u' - r u = k linear!
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