Mark44 said:If you wanted to annihilate, for example, etsin(3t) or etcos(3t), the annihilator would be D2 - 2D + 10.
The characteristic equation for the differential equation y'' - 2y' + 10y = 0, or (D2 - 2D + 10)y = 0, is r2 - 2r + 10 = 0. The roots of this latter equation are r = 1 +/- 3i. The connection between etsin(3t) and etcos(3t) on the one hand, and 1 +/- 3i on the other, is not coincidental.
The annihilator method is a technique used to solve ordinary differential equations (ODEs). It involves finding a differential operator, called the annihilator, that when applied to the given ODE, results in a simpler equation that can be easily solved.
The annihilator method is most useful when the ODE contains terms that can be eliminated by the application of a differential operator. It is particularly effective for solving non-homogeneous linear ODEs with constant coefficients.
To find the annihilator of an ODE, you must first identify the terms that need to be eliminated. Then, you apply the corresponding differential operator to those terms. For example, if the ODE contains a term in the form of e^x, you would apply the operator D-e^x to eliminate it.
No, the annihilator method is most effective for solving linear ODEs with constant coefficients. It may not work for nonlinear or variable coefficient ODEs.
No, there are various other methods for solving ODEs, such as separation of variables, substitution, and the method of undetermined coefficients. The annihilator method is just one approach that can be used, depending on the type of ODE.