Hermite's differential equation

Question

I'am asking the following question, because i couldn't find any credible, scientific resource which explicitly names the Hermite Differential equation as being linear.

So the question is: Is Hermite's Equation (link to definition: http://mathworld.wolfram.com/HermiteDifferentialEquation.html) a linear, homogenous ordinary differential equation?

My guess is that it is linear, because it is conform with the definition given for linear equations. (Link to linear definition: http://mathworld.wolfram.com/OrdinaryDifferentialEquation.html , equation (2))

Answer 1

The equation $$y''-2xy'+\lambda y=0$$ is

• linear because $y$ appears linearly; every term involving $y$ or its derivatives is of the form $a(x) y^{(k)}$.
• homogeneous because there are no terms not involving $y$. (Homogeneity is usually defined as a property of linear equations only.)

Answer 2

One can look at it like this: We have a differential operator $$ \frac{d^2}{dx^2} - 2x\frac{d}{dx} + \lambda. $$ Look at what happens if we apply it to a sum of two functions: $$ \left(\frac{d^2}{dx^2} - 2x\frac{d}{dx} + \lambda\right)(y_1+y_2). $$ This is equal to $$ \left(\frac{d^2}{dx^2} - 2x\frac{d}{dx} + \lambda\right)y_1 + \left(\frac{d^2}{dx^2} - 2x\frac{d}{dx} + \lambda\right)y_2. $$ And look at what happens when we apply it to the product of a function $y$ and a constant $c$: $$ \left(\frac{d^2}{dx^2} - 2x\frac{d}{dx} + \lambda\right)(cy). $$ We get $$ c\left(\left(\frac{d^2}{dx^2} - 2x\frac{d}{dx} + \lambda\right) y \right). $$ Therefore the differential operator is linear. Consequently the corresponding differential equation is linear.