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Thread: I need some help with this bernoulli's equation that is given in my assignment

  1. #1
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    I need some help with this bernoulli's equation that is given in my assignment

    solve y′−6y/x=y^5/x^13
    i have tried again and again but my answer is still wrong. can someone show me the steps to solving this question?
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  2. #2
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    Re: I need some help with this bernoulli's equation that is given in my assignment

    My attempt to solve it failed. I'm new to differentual equations. I saw this problem and thought that is easier than my homework. It turns out that I need help with this one, too. I don't know how you tried, but this is where things went wrong for me.

    It was not that obvious to me that this is a Bernoulli equation $y^′−\frac{6y}{x}=\frac{y^5}{x^{13}}$, so I did some rearrangements. $y^′=\frac{y^5}{x^{13}}+\frac{6y}{x}$ and $y^′=6yx^{-1}+y^5x^{-13}$ now I see it clearly.

    The first step from the algorithm I was taugh is to solve for the linear term.
    $\frac{\text{d}\bar{y}}{\text{d}x} = 6\bar{y}x^{-1}$
    $\bar{y}= Ce^{\int_{}^{}x^{-1}dx } = Ce^{ln(x)} = Cx$
    DONE.

    Now I'm looking for solutions of this form. $\bar{y}(x)=C(x)x$

    We have: $C(x)^′x+C(x)= 6C(x)+x^{-8}C(x)^5$ (if I'm not wrong)

    My teacher said that the e-terms ( C(x) and 6C(x) ) should always cancel each other, otherwise you made mistakes. I think I'm wrong, but I don't know where.
    Last edited by Hippasus; Oct 4th 2017 at 11:27 AM.
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  3. #3
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    Re: I need some help with this bernoulli's equation that is given in my assignment

    Take a look here: I need some help with this bernoulli's equation that is given in my assignment

    I think Hippasus is confusing Bernoulli equations with linear equations.
    Last edited by Archie; Oct 4th 2017 at 11:29 AM.
    Thanks from Hippasus
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  4. #4
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    Re: I need some help with this bernoulli's equation that is given in my assignment

    I learned an algorithm at university. The first step of solving a Bernoulli equation is to consider the linear equation associeted to it.

    Bernoulli's $\frac{\text{d}x}{\text{d}t}=a(t)x+b(t)x^\alpha, \alpha \in \Re\diagdown \left\{1,0\right\}$
    Linear $\frac{\text{d}x}{\text{d}t}=a(t)x$

    Am I mistaking ?
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  5. #5
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    Re: I need some help with this bernoulli's equation that is given in my assignment

    Doesn't look right to me. Although the usual method does turn the Bernoulli equation into a linear one.
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