# Third order differential equation

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.

Last edited:

BruceW
Homework Helper
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.

BruceW
Homework Helper
yep. looks good!