Undetermined coefficients vs. Variation of Parameters

[Sparky_]

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_

 

[nicksauce]

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.

 

[HallsofIvy]

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.