Identifying Types of Differential Equations: Tips for Beginners

[shaan_aragorn]

Hi all. I need some help here. I want to know, how can one differentiate or identify a type of a differential equation (like exact, variable separable form, reducible to variable separable form, homogenous, non-homogenous form). Please don’t suggest more practice. Are there any tricks to aid the identification?
Note: I am relatively new to DE.

 

[HallsofIvy]

The main trick is knowing the definitions!

Homogeneous equations should be identifiable directly from the definition.
(And, of course, if they don't satisfy that definition, then they are non-homogeneous!)

When you learned about exact equations, you should have learned the "cross-condition" for exactness: the differential equation f(x,y)dx+ g(x,y)dy= 0 is exact if and only if f_y= g_x. (I call that the "cross-condition" because f(x,y), the function multiplying dx, is differentiated wrt x and vice-versa.)

As for "separable" about the only way to determine that is to try to separate the variables! If the equation is separable, that should be pretty obvious.

Finally, reducible to separable: very easy to answer but you won't like my answer!
Theoretically, every first order d.e. has an "integrating factor" the reduces it to an exact equation and, theoretically, every exact equation can be made separable by a change of variable. So every first order d.e. can be "reduced to a separable d.e.". But there is no general way of finding either that integrating factor or the appropriate change of variable.