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.

 

[blue_raver22]

I can provide a response to the content regarding classifying ODE's. In order to classify an ODE, you are correct that you need to determine whether it is linear or non-linear and what order it is. However, there are specific criteria for determining linearity and order in ODE's.

First, a linear ODE is one where the dependent variable and its derivatives appear only in a linear form. This means that the terms in the ODE can only be multiplied by constants or the dependent variable and its derivatives. In contrast, a non-linear ODE contains terms that are not in a linear form, such as products, powers, or trigonometric functions.

Secondly, the order of an ODE is determined by the highest derivative present in the equation. For example, a first-order ODE contains only first derivatives, a second-order ODE contains second derivatives, and so on.

In the example provided, x2x'' + e-xx = t3, the highest derivative present is the second derivative, therefore this is a second-order ODE. Additionally, the presence of the term e-xx makes this a non-linear ODE.

In terms of homogeneity, this refers to whether all terms in the ODE contain the dependent variable and its derivatives. In the given example, dx/dt = logte-x, both terms contain the dependent variable t, therefore this can be classified as a homogeneous ODE.

In conclusion, when classifying ODE's, it is important to consider both linearity and order in order to accurately describe the equation. Additionally, the presence of the dependent variable in all terms can also determine whether the ODE is homogeneous or heterogeneous.