(Differential Equations) Determining Linearity of a Function

[electronicaneer]

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?

 

[Brian T]

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

 

[HallsofIvy]

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 $dy/dx= y^2$ is non-linear even though all of the derivatives are not to a power. And of course, you should include other, non-polynomial, functions. $dy/dx= sin(y)$ and $d^2y/dx^2+ e^{dy/dx}+ y= 0$ are non-linear differential equations.