Guessing - Judicious Guessing (or Undetermined...

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Judicious Guessing (or Undetermined Coefficients) Recall that Variation of Parameters gave us a way to produce a particular solution to the nonho- mogeneous equation, given that we could solve the homogeneous equation. Here we will look at a few shortcuts that will allow us to skip the sometimes difficult integrations required by Variation of Parameters. ± Judicious Guessing The goal is to find a particular solution ψ ( x ) to the equation y 00 + p ( x ) y 0 + q ( x ) y = g ( x ) by guessing the form of solution, and then solving for the undetermined coefficients in that form. For this reason this method is also known as the method of undetermined coefficients. It is not always applicable, but it does work in the following cases. Throughout, we will assume the homogeneous equation has constant coefficients. ± Case 1: RHS is polynomial In this case we will assume that our equation is of the form ay 00 + by 0 + cy = a 0 + a 1 x + ··· + a n x n . Here because the RHS is a polynomial, we will guess that the particular solution ψ ( x ) is also a polynomial. In order to make the degree of the polynomials on both sides agree, our guess will depend on which derivatives appear on the LHS. In this case we guess: ψ ( x ) = A 0 + A 1 x + ··· + A n x n if c 6 = 0 A 0 x + A 1 x 2 + ··· + A n x n +1 if c = 0 and b 6 = 0 A 0 x 2 + A 1 x 3 + ··· + A n x n +2 if c = 0 and b = 0 To find the undetermined coefficients A 0 ,A 1 ,...,A n , we plug in our guess and compare the coeffi- cients of t k for each k . Remark: We included the case where b = 0 and c = 0 for completeness, but you will NEVER NEED TO APPLY THIS CASE. Since the original equation in this case just looks like ay 00 = p ( x ) for some polynomial p ( x ) , we can find y just by integrating this polynomial twice.
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I Example Find a particular solution to the equation y 00 + 3 y 0 = x 2 + 6 x + 1 . Solution
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Guessing - Judicious Guessing (or Undetermined...

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