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Did I prove this Bernoulli equation correctly?

  1. Sep 16, 2006 #1

    Pengwuino

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    Gold Member

    Given a differential equation with the form:

    [tex]\frac{{dy}}{{dx}} + P(x)y = Q(x)y^n [/tex]

    and using the substitution [tex]v = y^{1 - n}[/tex]

    I attempted to prove that it transforms into

    [tex]\frac{{dv}}{{dx}} + (1 - n)P(x)v = (1 - n)Q(x)[/tex]

    Here’s the proof, did I do it correctly? I got the write answer so I assume I did :D

    [tex]\begin{array}{l}
    y = v^{ - 1 + n} \\
    \frac{{dv}}{{dx}} = \frac{{dv}}{{dy}}\frac{{dy}}{{dx}} \\
    \frac{{dv}}{{dy}} = (1 - n)y^{ - n} \frac{{dy}}{{dx}} \\
    \frac{{y^n }}{{(1 - n)}}\frac{{dv}}{{dx}} = \frac{{dy}}{{dx}} \\
    \frac{{y^n }}{{(1 - n)}}\frac{{dv}}{{dx}} + P(x)y = Q(x)y^n \\
    \frac{{dv}}{{dx}} + (1 - n)P(x)\frac{y}{{y^n }} = (1 - n)Q(x) \\
    \frac{{dv}}{{dx}} + (1 - n)P(x)y^{1 - n} = (1 - n)Q(x) \\
    \frac{{dv}}{{dx}} + (1 - n)P(x)v = (1 - n)Q(x) \\
    \end{array}
    [/tex]
     
    Last edited: Sep 16, 2006
  2. jcsd
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