MTH401 Final Term Short Notes

MTH401 Short Notes final term · Lec 19 to 45 · Website Link: · Important Note

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Lecture 19 Undetermined Coefficients: Annihilator Operator Approach

Undetermined Coefficients: Annihilator Operator Approach

The form of g(x): The input function g (x) has to have one of the following forms:

•  A constant function k.
•  A polynomial function
•  An exponential function e x
•  The trigonometric functions sin( x), cos( x)
•  Finite sums and products of these functions.

Otherwise, we cannot apply the method of undetermined coefficients.

Solution Method Consider the following non-homogeneous linear differential equation with constant coefficients of order n

If L denotes the following differential operator

Then the non-homogeneous linear differential equation of order n can be written as The function g(X) should consist of finite sums and products of the proper kind of functions as already explained.

Step 1 Write the given non-homogeneous linear differential equation in the form L( y)  g(x)

Step 2 Find the complementary solution yc by finding the general solution of the associated homogeneous differential equation:

L( y) = 0 Step 3 Operate on both sides of the non-homogeneous equation with a differential operator L1 that annihilates the function g(x).

Step 4 Find the general solution of the higher-order homogeneous differential equation L1L( y)  0

Step 5 Delete all those terms from the solution in step 4 that are duplicated in the complementary solution yc , found in step 2.

Step 6 Form a linear combination yp of the terms that remain. This is the form of a particular solution of the non-homogeneous differential equation L(y)  g(x)

Step 7 Substitute equation yp found in step 6 into the given non-homogeneous linear differential L(y)  g(x)

Match coefficients of various functions on each side of the equality and solve the resulting system of equations for the unknown coefficients in yp . Step 8 With the particular integral found in step 7, form the general solution of the given differential

equation as:

Example 1