Use of method of undetermined coefficients

In summary, the conversation discusses the use of "method of undetermined coefficients" to solve equations with a specific form of forcing function. The method is limited to certain types of functions, such as sinusoidal, polynomial, and exponential, due to the properties of linear operators with constant coefficients. For other types of forcing functions, alternative methods such as "variation of parameters" must be used. The conversation also touches on the possibility of using Taylor series expansion for large values of z and whether it can lead to a series solution.
  • #1
vineethbs
8
0
Use of "method of undetermined coefficients"

Hi all,

Suppose I have a equation

[tex]f(z+1) - f(z) = z^{1/2} , \forall z \geq 0 [/tex]eq (1)

then is it possible to solve this equation by the method of undetermined coefficients ?

It is usually seen in textbooks that the forcing function is taken to be sinusoidal or polynomial or exponential when the method of undetermined coefficients is used. Why is this so ? What kind of properties must the forcing function satisfy so that this method can be used ?

In the above eq (1), if suppose I assume that f(z) is say
[tex]c_{1} z^{1/2} + c_{2} + c_{3} z^{-1/2} + \cdots[/tex]
and then substitute in eq (1),
then can I do a Taylor series expansion for an arbitrarily large z ?
for eg :
[tex]c_{1} (z + 1)^{1/2} = c_{1} z^{1/2} * (1 + \frac{1}{z})^{1/2}[/tex]

and then expand out the 1 + .. term in a Taylor series expansion ?

Does this lead to a series solution ?Thanks in advance for spending your time on this
 
Last edited:
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  • #2


vineethbs said:
Hi all,

Suppose I have a equation

[tex]f(z+1) - f(z) = z^{1/2} , \forall z \geq 0 [/tex]eq (1)

then is it possible to solve this equation by the method of undetermined coefficients ?

It is usually seen in textbooks that the forcing function is taken to be sinusoidal or polynomial or exponential when the method of undetermined coefficients is used. Why is this so ? What kind of properties must the forcing function satisfy so that this method can be used ?
Essentially because those are the kind of solution you would expect for a linear difference (or differential) equation with constant coefficients because a linear operator with constant coefficients will map those kinds of functions into themselves. If the "forcing function" (more physics terminology- Bah!) is not of that kind you cannot use the "method of undetermined coefficients" because you cannot expect the correct function to be a square root. You will need to use something like "variation of parameters".

In the above eq (1), if suppose I assume that f(z) is say
[tex]c_{1} z^{1/2} + c_{2} + c_{3} z^{-1/2} + \cdots[/tex]
and then substitute in eq (1),
then can I do a Taylor series expansion for an arbitrarily large z ?
for eg :
[tex]c_{1} (z + 1)^{1/2} = c_{1} z^{1/2} * (1 + \frac{1}{z})^{1/2}[/tex]

and then expand out the 1 + .. term in a Taylor series expansion ?

Does this lead to a series solution ?


Thanks in advance for spending your time on this
 

1. What is the method of undetermined coefficients?

The method of undetermined coefficients is a technique used in mathematics and physics to solve differential equations. It involves finding a particular solution to a non-homogeneous differential equation by assuming a general form for the solution and then solving for the coefficients.

2. When is the method of undetermined coefficients used?

This method is typically used when the differential equation has a non-homogeneous term, such as a constant or a function, on the right-hand side. It is also used when the equation has a polynomial, exponential, or trigonometric function as a solution.

3. How does the method of undetermined coefficients work?

The method of undetermined coefficients works by assuming a general form for the particular solution, which includes undetermined coefficients. These coefficients are then solved for by substituting the assumed solution into the original differential equation. The particular solution is then found by plugging in these coefficients to the assumed solution.

4. What are the limitations of the method of undetermined coefficients?

The method of undetermined coefficients may not work for all types of non-homogeneous differential equations. It is most effective for equations with simple non-homogeneous terms and solutions. Additionally, it may not work for equations with repeated roots or for equations with non-constant coefficients.

5. Are there any alternative methods to solve non-homogeneous differential equations?

Yes, there are other methods such as variation of parameters and the Laplace transform method. These methods may be more effective for certain types of non-homogeneous equations and can be used as alternatives to the method of undetermined coefficients.

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