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Guess_Method

# Guess_Method - 21 Method of Undetermined Coefcients(aka...

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21 Method of Undetermined Coefficients (aka: Method of Educated Guess) In this chapter, we will discuss one particularly simple-minded, yet often effective, method for finding particular solutions to nonhomogeneous differential equations. As the above title suggests, the method is based on making “good guesses” regarding these particular solutions. And, as always, “good guessing” is usually aided by a thorough understanding of the problem (the ‘education’), and usually works best if the problem is simple enough. Fortunately, you have had the necessary education, and a great many nonhomogeneous differential equations of interest are sufficiently simple. As usual, we will start with second-order equations, and then observe that everything devel- oped also applies, with little modification, to similar nonhomogeneous differential equations of any order. 21.1 Basic Ideas Suppose we wish to find a particular solution to a nonhomogeneous second-order differential equation ay ′′ + by + cy = g . If g is a relatively simple function and the coefficients — a , b and c — are constants, then, after recalling what the derivatives of various basic functions look like, we might be able to make a good guess as to what sort of function y p ( x ) yields g ( x ) after being plugged into the left side of the above equation. Typically, we won’t be able to guess exactly what y p ( x ) should be, but we can often guess a formula for y p ( x ) involving specific functions and some constants that can then be determined by plugging the guessed formula for y p ( x ) into the differential equation and solving the resulting algebraic equation(s) for those constants (provided the initial ‘guess’ was good). ! Example 21.1: Consider y ′′ 2 y 3 y = 36 e 5 x . 417

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418 Method of Educated Guess Since all derivatives of e 5 x equal some constant multiple of e 5 x , it should be clear that, if we let y ( x ) = some multiple of e 5 x , then y ′′ 2 y 3 y = some other multiple of e 5 x . So let us let A be some constant “to be determined”, and try y p ( x ) = Ae 5 x as a particular solution to our differential equation: y p ′′ 2 y p 3 y p = 36 e 5 x equal1⇒ bracketleftbig Ae 5 x bracketrightbig ′′ 2 bracketleftbig Ae 5 x bracketrightbig 3 bracketleftbig Ae 5 x bracketrightbig = 36 e 5 x equal1⇒ bracketleftbig 25 Ae 5 x bracketrightbig 2 bracketleftbig 5 Ae 5 x bracketrightbig 3 bracketleftbig Ae 5 x bracketrightbig = 36 e 5 x equal1⇒ 25 Ae 5 x 10 Ae 5 x 3 Ae 5 x = 36 e 5 x equal1⇒ 12 Ae 5 x = 36 e 5 x equal1⇒ A = 3 . So our “guess”, y p ( x ) = Ae 5 x , satisfies the differential equation only if A = 3 . Thus, y p ( x ) = 3 e 5 x is a particular solution to our nonhomogeneous differential equation. In the next section, we will determine the appropriate “first guesses” for particular solutions corresponding to different choices of g in our differential equation. These guesses will involve specific functions and initially unknown constants that can be determined as we determined A in the last example. Unfortunately, as we will see, the first guesses will sometimes fail. So we
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Guess_Method - 21 Method of Undetermined Coefcients(aka...

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