Hermite's differential equation 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)) 
 A: 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.)

A: 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.
