# Third order differential equation

• electronic engineer
In summary, the conversation was about understanding differential equations and specifically moving from second order to third order. The principle used was to apply the product rule, as shown in the equations provided, and the final result was obtained by applying the product rule to the third order 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.

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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).

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

which leads to the final result.

yep. looks good!