# Question on checking the linearity of a differential operator

1. Dec 12, 2013

### Seydlitz

Suppose I have this operator:

$D^2+2D+1$.

Is the $1$ there, when applied to a function, considered as identity operator?

Say:

$f(x)=x$.

Applying the operator results in:

$D^2(x)+2D(x)+(x)$ or $D^2(x)+2D(x)+1$?

If $1$ here is considered as an identity operator then the answer will be the former, and the whole operator is linear. But if it's the former, I don't see why the operator will be linear because of the extra $1$ term.

$D$ refers to $\frac{d}{dx}$, and the power of it refers to the order of the derivative.

2. Dec 13, 2013

### tiny-tim

Hey Seydlitz!

If D = d/dx, then $(D^2+2D+1)(y) = y'' + 2y' + y$

3. Dec 13, 2013

### Seydlitz

Ok thanks tiny-tim!