Classifying ODE's: What Determines Linearity and Order?

[Lengalicious]

When classifying an ODE do I just say whether its linear / non-linear and what order it is?

EDIT: Example: x²x'' + e^-xx = t³ where x is function of t.

is this 2nd order, non linear and heterogeneous?

Also, would dx/dt = logte^-x be homogeneous because both terms contain the dependant variable 't'?

 

[bigfooted]

There are some more classifications for (second order) odes, mostly based on the fact that the ode is solvable (or not) when it is of a certain class. I like the choice of the Maple software, which is pretty classic. The online description of odeadvisor giving you a classification is here:
http://www.maplesoft.com/support/help/Maple/view.aspx?path=DEtools/odeadvisor

Your first order ode is homogeneous, because it does not have a term that only depends on t. dx/dt=a(t)*x+b(t) is not homogeneous, but dx/dt = a(t)*x is. Your example is also separable, which means it can be solved using separation of variables.

 

[Lengalicious]

so if dx/dt = logte^-1, would that now mean that this was no longer homogeneous? Thanks for the help by the way.

 

[alan2]

Your equation is not homogeneous because of the t cubed term.