These properties are of practical and theoretical importance because they enable us to obtain new solutions from given ones. Thus in modeling, whenever possible, we prefer linear ODEs over nonlinear ones, which have no similar properties.

Show that nonhomogeneous linear ODEs (1) and homogeneous linear ODEs (2) have the following properties. Illustrate each property by a calculation for two or three equations of your choice. Give proofs.

y = 0 (that is, y(x) = 0 for all x, also written y(x) = 0) is a solution of (2) [not of (1) if r(x) ≠ 0!], called the trivial solution.