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!

 

[blue_raver22]

I can understand your confusion with the transition from second order to third order differential equations. Let me explain it to you in a simpler way.

A differential equation is an equation that relates a function with its derivatives. The order of a differential equation is determined by the highest order derivative present in the equation. So, a third order differential equation is one that involves the third derivative of the function.

In the given example, we have a function y and its first and second derivatives, represented as y' and y''. The third derivative, y''', is obtained by taking the derivative of y'' with respect to x. This is known as the chain rule in calculus.

To simplify the notation, we use the shorthand notation of partial derivatives, where $$f_{x}$$ represents the partial derivative of f with respect to x. Applying this notation, we can rewrite the third derivative as y''' = $$f_{xx}+2ff_{xy}+f_{yy}f^{2}+f_{x}f_{y}+ff_{y}^{2}$$.

I hope this helps you understand the transition from second order to third order differential equations. Keep exploring and learning, the more you practice, the clearer it will become. Good luck!