Differential Equations 6

Differential operators

Let \(D=\frac{d}{dx}\), \(D^2=\frac{d^2}{dx^2}\), and so on

Let \(f(D)\) is a polynomial on \(D\)

\(f(D)y\) is a linear homogeneous differential function

Let \(y=e^{mx}\), \(D^ke^{mx} = m^ke^{mx}\)

Let \(y=e^{mx}\), \(f(D)e^{mx} = e^{mx}f(m)\)

\((D - m)^n\) is a polynomial.

Let operator be \((D - m)^n\) and \(y=x^ke^{mx}\), \((D - m)^k x^ke^{mx} = k!e^{mx}\)

\((D - m)^n x^ke^{mx} = 0\) for \(k = 0, 1, 2, .... , (n - 1)\)