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.

 

[tacman]

Hello there,

Identifying the type of a differential equation can seem daunting at first, but with some practice and understanding of the characteristics of each type, it can become easier. Here are some tips that can help you in identifying the type of a differential equation:

1. Look at the highest order of the derivative present in the equation. This will give you a clue about the type of the equation. For example, if the highest order is 1, it could be a first-order equation, and if it is 2, it could be a second-order equation.

2. Check if the equation is separable. This means that the equation can be written as a product of two functions, one depending on the independent variable and the other on the dependent variable. If this is the case, then it is a separable differential equation.

3. Look for any constant coefficients in the equation. If the coefficients are constants, then it could be a linear differential equation. If the coefficients are functions of the independent variable, then it could be a non-linear equation.

4. Check if the equation is homogeneous. A homogeneous equation is one in which all the terms have the same degree of the dependent variable. This means that if you replace the dependent variable with a constant multiple of itself, the equation remains the same. If this is the case, then it is a homogeneous differential equation.

5. Determine if the equation is exact. An exact differential equation is one in which the total differential of the equation can be written as a linear combination of the variables and their differentials. This means that if you take the partial derivatives of the equation with respect to the variables, they will be equal. If this is the case, then it is an exact differential equation.

These are just some general tips that can help you identify the type of a differential equation. However, it is important to note that there is no one specific trick that can work for all equations. Practice and familiarity with the different types of equations will ultimately help you in identifying them more easily.

I hope this helps. Good luck with your studies!