What does it mean to be a linear ODE?

What does it mean to be a linear ODE?#

Take two solutions \(X_1\) and \(X_2\) with the following initial conditions and inputs, respectively:

\[ (X_{1,0}, u_1) \quad \text{for } X_1, \qquad (X_{2,0}, u_2) \quad \text{for } X_2. \]

Then,

\[ X(t) = \alpha X_1(t) + \beta X_2(t) \]

is a solution with the initial condition

\[ \alpha X_{1,0} + \beta X_{2,0} \]

and input

\[ \alpha u_1 + \beta u_2, \]

for any scalars \(\alpha\) and \(\beta.\)

This property of linear systems helps construct the solution as an appropriate combination of simpler solutions.

What are some of the ways in which we will use this property?