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Third order differential equation

  1. Jan 16, 2015 #1
    Hi all, I need to understand these differential equations specially moving from the second order to the third order because i couldn't understand how they got to the result, what was exactly the principle:

    $$ y'=f(x,y) $$

    $$ y''=\frac{df}{dx}(x,y(x)) = f_{x}(x,y) + f_{y}(x,y)y' = f_{x}(x,y) + f_{y}(x,y)f(x,y) $$

    $$ y'''=f_{xx}+2ff_{xy}+f_{yy}f^{2}+f_{x}f_{y}+ff_{y}^{2} $$

    where $$ f_{x} $$ is the partial derivation of x and so for the similar other quantities.
    please help me with it, thank you.
     
    Last edited: Jan 16, 2015
  2. jcsd
  3. Jan 16, 2015 #2

    BruceW

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    use the same principles you used for going from first order to second order. You will need to use product rule, because ##y''## contains ##f_yf## which is a product of two functions. But it is not much more complicated than going from first order to second order.

    hint: for any function ##g(x,y)## you have: ##g'=g_x+g_y y'## (where ##g'## means total derivative with respect to x).
     
  4. Jan 16, 2015 #3
    $$ y''' = f_{xx} + f_{xy} y' + f ( f_{yx} + f_{yy} f) + f_{y} (f_{x} + f_{y} f) $$

    which leads to the final result.
     
  5. Jan 17, 2015 #4

    BruceW

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    yep. looks good!
     
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