1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

(Differential Equations) Determining Linearity of a Function

  1. Feb 13, 2016 #1
    This is more of a general question than a specific homework question, because it popped up in more than 1 problem. If you have 'x' has the independent variable and 'y' as the dependent variable, you can determine the linearity in 'y' by seeing if any of the derivatives (dy/dx) are being raised to a power or not.

    How would you go about determining the linearity in 'x' then though? Would you rearrange it so all of the derivatives reflect (dx/dy) and then re-evaulaute?
     
  2. jcsd
  3. Feb 13, 2016 #2
    I suppose you could but generally, unless you're talking about first order equations, it would be hard and pointless to do. For example, think of a DE which gives a particles trajectory x as a function of time t. In this scenario, it would be perfectly fine to have two different times correspond to one position. However upon flipping it, and trying to make t as a function of x, you would have one position corresponding to two different times (this tells you that t really is not the dependent variable) . The example shows that although sometimes you can (in simple cases), it might produce mathematically inconsistencies or just be impossible to do analytically

    Also, as a side note, what you're trying to determine in differential eq is the linearity of the differential operator on the solution (y) and not the linearity in y itself
     
  4. Feb 14, 2016 #3

    HallsofIvy

    User Avatar
    Staff Emeritus
    Science Advisor

    There is no reason to define "linear in x" for a differential equation because it has no effect on how the equation might be solved. A "linear differential equation" is one that is "linear in y". By the way, you say "you can determine the linearity in 'y' by seeing if any of the derivatives (dy/dx) are being raised to a power or not." I presume you intended to include y itself in "any of the derivatives". The equation [itex]dy/dx= y^2[/itex] is non-linear even though all of the derivatives are not to a power. And of course, you should include other, non-polynomial, functions. [itex]dy/dx= sin(y)[/itex] and [itex]d^2y/dx^2+ e^{dy/dx}+ y= 0[/itex] are non-linear differential equations.
     
    Last edited by a moderator: Feb 14, 2016
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted



Similar Discussions: (Differential Equations) Determining Linearity of a Function
Loading...