MHB Determining Order of Differential Equations

[ineedhelpnow]

Hello. I can't seem to remember how to do these kind of problems. I need to determine the order of the differential equations. Can someone show how this is done so that I can understand how to do the rest?

$\d{^2y}{x^2}+2\d{y}{x} \d{^3y}{x^3}+x=0$

 

[Ackbach]

The order of a DE is the order of the highest derivative present in the DE. What order is that?

 

[ineedhelpnow]

Three. Got it. Thanks.

 

[chisigma]

Hello. I can't seem to remember how to do these kind of problems. I need to determine the order of the differential equations. Can someone show how this is done so that I can understand how to do the rest?

$\d{^2y}{x^2}+2\d{y}{x} \d{^3y}{x^3}+x=0$

The DE is of third order, of course... but with the substitution $\displaystyle y^{'} = u$ it becomes...

$\displaystyle u^{\ '} + 2\ u\ u^{\ ''} + x = 0\ (1)$

... which is of order two... solving (1) however is a different and not necessarly trivial task...

Kind regards

$\chi$ $\sigma$

 

[Ackbach]

Three. Got it. Thanks.

You got it! Although:

The DE is of third order, of course... but with the substitution $\displaystyle y^{'} = u$ it becomes...

$\displaystyle u^{\ '} + 2\ u\ u^{\ ''} + x = 0\ (1)$

... which is of order two... solving (1) however is a different and not necessarly trivial task...

Kind regards

$\chi$ $\sigma$

You see what chisigma is getting at? Essentially, you can reduce the order of the DE with a substitution. This is generally possible when the function itself, $y$, is not present in the DE.