Undetermined coefficients vs. Variation of Parameters

In summary, undetermined coefficients and variation of parameters are two procedures that can be used to solve non-homogeneous linear equations. However, undetermined coefficients only works if the inhomogeneous term is an exponential, sine/cosine, or a combination of them, while Variation of Parameters always works, but the math may be more complicated. It is important to note that undetermined coefficients may not work for all types of non-homogeneous terms, while variation of parameters can be applied to any type. Ultimately, the decision of which method to use depends on the given equation and its specific terms.
  • #1
Sparky_
227
5
Greetings,

Regarding the two procedures: undetermined coefficients and variation of parameters, can both procedures be used interchangeably - meaning they both solve (non-homogeneous linear equations)?

Does one method work better in certain situations, if so which method is preferred when?

How can one know when to use which method?

Thanks
-Sparky_
 
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  • #2
If I recall correctly, undetermined coefficients only works if the inhomogeneous term is an exponential, sine/cosine, or a combination of them, while Variation of Parameters always works, but the math is a little more messy.
 
  • #3
nicksause is correct. The "possible solutions" to a linear equation with constant coefficients must be: exponential, polynomial (he forgot those!), sine or cosine, or combinations of those. "Undetermined Coefficients" only works if the right-hand side of the equation is one of those.

For example, y"+ y= ln(x) or y"- 2y'+ y= tan(x) cannot be done by undetermined coefficients. They can be solved by variation of parameters- though you might not be able to do the resulting integral.
 

1. What is the difference between undetermined coefficients and variation of parameters?

Undetermined coefficients and variation of parameters are two methods used to solve differential equations. Undetermined coefficients is a method where a particular solution is assumed based on the form of the non-homogeneous term in the differential equation, while variation of parameters is a method where the particular solution is found by integrating a general solution to the associated homogeneous equation.

2. Which method should I use to solve a non-homogeneous differential equation?

The choice between undetermined coefficients and variation of parameters depends on the form of the non-homogeneous term. If the non-homogeneous term is a polynomial, exponential, or trigonometric function, undetermined coefficients is often the faster and easier method. However, if the non-homogeneous term is a more complicated function, variation of parameters may be a better approach.

3. Can undetermined coefficients and variation of parameters be used to solve any type of differential equation?

No, these methods can only be used for linear differential equations with constant coefficients. They cannot be used for non-linear equations or equations with variable coefficients.

4. How do I know if my initial conditions are satisfied when using undetermined coefficients or variation of parameters?

When using undetermined coefficients, the particular solution will always satisfy the non-homogeneous term, but may not satisfy the initial conditions. When using variation of parameters, the particular solution will satisfy both the non-homogeneous term and the initial conditions.

5. Are there any situations where one method is better than the other?

In general, undetermined coefficients is a faster and easier method, but it may not work for more complicated non-homogeneous terms. Variation of parameters may take longer, but it can be used for a wider range of non-homogeneous terms. It is best to try both methods and see which one works best for a particular differential equation.

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