Linearity (Differential Equations)

How do we know an equation is linear in y or linear in x?

For example... if the equation is x^2y" + 4xy' - 16y = 0

It says this is linear in y.

The other example, if the equation is xy" - (3x + y)y' = 14x

This one is linear in x. I don't know why.

The other example is y"/x + 36y' - e^y = 0

This equation is neither linear in y nor linear in x.

Can sombody explain about this for me please...? T_____T

linear in y: looking at the equation term-by-term, there is only one y-term, and it can either be a derivative (of any order) or y itself, but cannot be any terms involving y^n, where n is greater than 1, i.e. no terms like y^2, or y^5, nor can it be embedded in functions like sin(y), ln(y), etc. Such terms are called nonlinear terms. Here it is noted that "mixed" terms, like y(dy/dx) are also mixed terms, and are nonlinear.

Your first equation is linear in y, y terms are on their own, they involve simple derivatives

Your second equation is not linear in y because it has a mixed term: yy' = y (dy/dx)

your third equation is not linear in y because it has y inside a function: e^y



linear in x: it means only "linear" terms appear, a linear term is of the form mx + b

Your first equation is nonlinear in x, x^2 is not linear

Your second equation is linear in x, only linear terms (x and constants) appear

Your second equation is nonlinear in x, terms like 1/x = x^{-1} are not linear.

I segmented linear in x and y above, but they are the same things, meaning they contain combinations of mx + b or my + b only (which are linear), you do not have to memorize what "mixed" terms are, etc., you just need to see which terms have a power of y^2 or x^2 or greater.

Note that any derivative is single order in y, i.e. say you have a length scale L, a derivative dy/dx ~ y / L, d^2y/dx^2 ~ y / L^2, d^5 y / dx^5 = y / L^5

so you can see that terms like y(dy/dx) ~ y^2 which is not linear, y^3(d^7 y/dx^7) ~ y^10, nonlinear, and functions like e^y are not linear either. Just look out for combinations that are of order x^2 or larger or y^2 or larger.

To test if the linear equation is linear in y, treat x as a constant and not a function. Then it is linear if the equation only contains y and the derivatives of y in a linear combination. That means you can write it as

a0(x) + a1(x) y + a2(x) y' + a3(x) y'' + a4(x) y''' ... = 0

In your second example, you have a factor y * y', so that's not a linear combination and therefore the equation is not linear in y.

In the third equation, y enters as e^y, so it's not a linear combination.

It's also not linear in x, because x enters as 1/x, which isn't a linear function.