Method of Undetermined Coefficients or Guessing Method

As for the second order case, we have to satisfy two conditions. One is already satisfied since we assumed that our equation has constant coefficients. The second condition has to do with the nonhomogeneous term g(x). Indeed, in order to use the undetermined coefficients method, g(x) should be one of the elementary forms

displaymath313,

where tex2html_wrap_inline315 is a polynomial function. For a more general case, see the remark below. In order to guess the form of the particular solution we follow these steps:

Remark: The undetermined coefficients method can still be used if

displaymath355,

where tex2html_wrap_inline357 has the elementary form described above. Indeed, we need (as we did for the second order case) to split the equation (NH) into m equations. Find the particular solution to each one, then add them to generate the particular solution of the original equation.

Example: Find a particular solution of

displaymath363

Solution: Let us follow these steps:

(1)
Characteristic equation

displaymath365

We have the factorization tex2html_wrap_inline367 . Therefore, the roots are 0,2,-2 and they are all simple.

(2)
We have to split the equation into the following two equations:

displaymath371;

(3)
The particular solution to the equation (1):
(3.1)
We have tex2html_wrap_inline373 which is a simple root. Then s = 1;
(3.2)
The guessed form for the particular solution is

displaymath377,

where A and B are to be determined. We will omit the detail of the calculations. We get A = -1/8 and B=0. Therefore, we have

displaymath387;

(4)
The particular solution to the equation (2):
(4.1)
We have tex2html_wrap_inline389 which is not a root. Then s = 0;
(4.2)
The guessed form for the particular solution is

displaymath393

where A and B are to be determined. We will omit the detail of the calculations. We get A = 0 and B=-3/5. Therefore, we have

displaymath403;

(5)
The particular solution to the original equation is given by

displaymath405

[Differential Equations] [First Order D.E.] [Second Order D.E.]
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Author: Mohamed Amine Khamsi

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