# Third order differential equation

## Main Question or Discussion Point

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:

Related Differential Equations News on Phys.org
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!