1.1 Linear DOs

A differential operator $L$ is linear if it satisfies the following, that is

• Additivity: $L[X+Y]=L[X]+L[Y]$
• Homogeneity of Order 1: $L[cx]=aL[x]$

Note that when the DO is linear the $\cdot$ as a placeholder for the argument is usually omitted, as every term must multiply it anyway

\frac{d^2}{dt^2}+g(t)$$ "Derivatives are linear; anything that combines derivatives with multiplication will be linear"