October 4 Math 2306 sec. 51 Fall 2019

Section 9: Method of Undetermined Coefficients

The context here is linear, constant coefficient, nonhomogeneousequations

any (n) + an−1y (n−1) + · · ·+ a0y = g(x)

where g comes from the restricted classes of functionsI polynomials,I exponentials,I sines and/or cosines,I and products and sums of the above kinds of functions

Recall y = yc + yp, so we’ll have to find both the complementary andthe particular solutions!

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Motivating ExampleFind a particular solution of the ODE

y ′′ − 4y ′ + 4y = 8x + 1

Question: What kind of function yp(x) when plugged into the rightside might equal a first degree polynomial? Perhaps another firstdegree polynomial.

Make an educated guess that

yp(x) = Ax + B for some numbers A and B.

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The Method: Assume yp has the same form as g(x)

y ′′ − 4y ′ + 4y = 6e−3x

Question: What kind of function yp(x) when plugged into the rightside might equal a multiple of e−3x?

Make an educated guess that

yp(x) = Ae−3x for some number A.

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Make the form general

y ′′ − 4y ′ + 4y = 16x2

There are two ways to view the right side g(x) = 16x2, as a monomialAx2 or as a quadratic expression. Let’s try assuming that

yp(x) = Ax2

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General Form: sines and cosines

y ′′ − y ′ = 20 sin(2x)

If we assume that yp = A sin(2x), taking two derivatives would lead tothe equation

−4A sin(2x)− 2A cos(2x) = 20 sin(2x).

This would require (matching coefficients of sines and cosines)

−4A = 20 and − 2A = 0.

This is impossible as it would require −5 = 0!

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General Form: sines and cosines

We must think of our equation y ′′ − y ′ = 20 sin(2x) as

y ′′ − y ′ = 20 sin(2x) + 0 cos(2x).

The correct format for yp is

yp = A sin(2x) + B cos(2x).

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Examples of Forms of yp based on g (Trial Guesses)

(a) g(x) = 1 (constant function)

(b) g(x) = x − 7 (1st degree polynomial)

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Examples of Forms of yp based on g (Trial Guesses)

(c) g(x) = 5x2 (2nd degree polynomial)

(d) g(x) = 3x3 − 5 (3rd degree polynomial)

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Examples of Forms of yp based on g (Trial Guesses)

(e) g(x) = xe3x (1st degree polynomial times e3x )

(f) g(x) = cos(7x) (sum of cos(7x) and sin(7x))

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Examples of Forms of yp based on g (Trial Guesses)

(g) g(x) = sin(2x)− cos(4x)(sum of cos(2x), sin(2x), cos(4x) and sin(4x))

(h) g(x) = x2 sin(3x)(2nd degree poly. times sin(3x) and 2nd degree poly. times cos(3x))

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Examples of Forms of yp based on g (Trial Guesses)

(i) g(x) = ex cos(2x) (sum of ex cos(2x) and ex sin(2x))

(j) g(x) = xe−x sin(πx)(sum of 1st deg. poly. times ex cos(πx) and 1st deg. poly. timesex sin(πx))

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