Linear Differential Operator
An operator is linear when for constants : the principle of superposition.
Definition 6.1 (Linear Differential Operator)
A differential operator is linear if for any functions and constants ,
which is called the principle of superposition.
Solving by superposition
Linearity of splits the solution of into two parts:
- Find the complementary functions satisfying the homogeneous equation .
- Find a particular integral satisfying .
- Combine them: since , the sum solves the full equation.
A 2nd order ODE has two linearly independent complementary functions, so the general solution of the full equation is
Related
Stated in
- Definition 6.1 (Linear Differential Operator)§6.1 2nd Order ODEs with Constant Coefficients