Third order differential equation

[electronic engineer]

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.

 

[BruceW]

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).

 

[electronic engineer]

$$ y''' = f_{xx} + f_{xy} y' + f ( f_{yx} + f_{yy} f) + f_{y} (f_{x} + f_{y} f) $$

which leads to the final result.

 

[BruceW]

yep. looks good!