1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: Did I prove this Bernoulli equation correctly?

  1. Sep 16, 2006 #1

    Pengwuino

    User Avatar
    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
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook

Can you offer guidance or do you also need help?
Draft saved Draft deleted