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Math Help - Solving Bernoulli Differential Equation.

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    Question Solving Bernoulli Differential Equation.

    Last edited by mr fantastic; January 2nd 2012 at 05:45 PM. Reason: Title.
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    Re: NEED TO SOLVE >>> bernoulli equation

    Quote Originally Posted by amro05 View Post
    Start by making the substitution \displaystyle \begin{align*} v = y^{1-n} \end{align*}.
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    Re: NEED TO SOLVE >>> bernoulli equation

    Quote Originally Posted by prove it View Post
    start by making the substitution \displaystyle \begin{align*} v = y^{1-n} \end{align*}.
    yes sir i did it

    but the problem is the absulet number (( 2 )) << without it , it became easy
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    Re: NEED TO SOLVE >>> bernoulli equation

    Quote Originally Posted by amro05 View Post
    Applying the 'standard procedure' the substitution w=\frac{1}{y} leads to the DE...

    w^{'}= - 1+2 w x -2 w^{2} (1)

    ... which is known as 'Riccati Equation'. The solution of this type of equation is usually an hard task but, may be, not impossible in this case. The question requires further investigation...

    Kind regards

    \chi \sigma
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    Re: NEED TO SOLVE >>> bernoulli equation

    Quote Originally Posted by chisigma View Post
    Applying the 'standard procedure' the substitution w=\frac{1}{y} leads to the DE...

    w^{'}= - 1+2 w x -2 w^{2} (1)

    ... which is known as 'Riccati Equation'. The solution of this type of equation is usually an hard task but, may be, not impossible in this case. The question requires further investigation...

    Kind regards

    \chi \sigma
    thank you sir for your help

    I begin with


    and I start with y1=2 and with y1=2x >>> but both gives more complicate Eq.

    thank you sir for your help
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    MHF Contributor chisigma's Avatar
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    Re: NEED TO SOLVE >>> bernoulli equation

    Quote Originally Posted by amro05 View Post
    The original DE...

    y^{'}=2-2 x y +y^{2} (1)

    ... is itself a 'Riccati Equation' so that a substitution of the type w=\frac{1}{y} is useless. More promising is the substiution w=y-x after that the (1) becomes...

    w^{'}=1-x^{2} + w^{2} (2)

    The (2) is also a 'Riccati Equation' but, very important detail, we can verify that w_{1}=x is solution of (2) and that allows You to find the general solution as follows. Setting w=w_{1}+u=x+u the (2) becomes...

    u^{'}=2 u x + u^{2} (3)

    ... which is a 'Bernoulli Equation', the solution of which is left to You...

    Kind regards

    \chi \sigma
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    Re: NEED TO SOLVE >>> bernoulli equation

    Quote Originally Posted by chisigma View Post
    Kind regards

    \chi \sigma

    Thank you very very mach .
    all the best

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