1. Not finding help here? Sign up for a free 30min 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!

Differential equation substitution method

  1. May 18, 2009 #1

    tony873004

    User Avatar
    Science Advisor
    Gold Member

    [tex]\begin{array}{l}
    \frac{{dy}}{{dx}} = \frac{{4x^2 + 5xy + y^2 }}{{x^2 }} \\
    \\
    \frac{{dy}}{{dx}} = 4 + \frac{{5y}}{x} + \left( {\frac{y}{x}} \right)^2 \\
    \\
    {\rm{Let }}v = y/x\,\,\,\, \Rightarrow \,\,\,\,y = vx \\
    \\
    \frac{{dy}}{{dx}} = 4 + \frac{{5v}}{{xx}} + \left( {\frac{{vx}}{x}} \right)^2 \\
    \\
    \frac{{dy}}{{dx}} = 4 + \frac{{5v}}{{x^2 }} + v^2 \\
    \end{array}[/tex]

    But the class notes say the final line should be
    [tex]\frac{dy}{dx}=x\frac{dv}{dx}+v[/tex]
    How did he get that?
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. May 18, 2009 #2
    Take the derivative of y = vx with respect to x, then equate it, and you are done.
     
  4. May 18, 2009 #3

    gabbagabbahey

    User Avatar
    Homework Helper
    Gold Member

    This should be [itex]\frac{dy}{dx}= 4 + \frac{{5vx}}{{x}} + \left( {\frac{{vx}}{x}} \right)^2=4+5v+v^2[/itex]

    Use the product rule to differentiate [itex]y=vx[/itex].
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook