Differential Equations Sanchez




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Differential Equations Sanchez

Part II -Summary 6

Annihilators
Annihilators. An annihilator of a function y=f(t) is a linear differential P(D) that satisfies the condition P(D)[f(t)] =0. This is the same as saying that f(t) is a solution of the homogenous differential equation P(D)f(t)=0.
Example: the homogeneous solution of the linear differential equation


Problem 1. Find an annihilator for each of the following functions:

Function

Annihilator






2.D-3
















6. (D-1)(D+2)

7. 2t +5














































Finding the general form of the particular solution of a non-homogeneous linear differential equation by using annihilators.


Step 1. Express the DE in linear differential form, that is, L(y)=g(x)
Step 2. Find the homogeneous solution (complementary solution) of the differential equation, that is find the general solution of L(y)=0
Step 3.. Find an annihilator L1for g(x), that is L1(g(x)=0
Step 4. Operate on both sides of the non-homogeneous equation with the annihilator L1, that is,


Problem 2. Find the general form of a particular solution of the following differential equation.

















.

Problem 4.

Solve the given differential equation by un-determinate coefficients




Problem 5.



Solve the DE of problem 3 by using the exponential shifting.




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