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# lecture_19 - MA 36600 LECTURE NOTES MONDAY MARCH 2 Method...

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MA 36600 LECTURE NOTES: MONDAY, MARCH 2 Method of Undetermined Coefficients Undetermined Coefficients. Say that we wish to solve a constant coefficient linear second order differ- ential equation in the form a y 00 + b y 0 + c y = f ( t ) We know how to find the general solution y = y ( t ) once we find homogeneous solutions y 1 = y 1 ( t ) and y 2 = y 2 ( t ), so we explain a method to find a particular solution Y = Y ( t ). Follow these three steps: #1. Express the function on the right-hand side as the sum of functions f ( t ) = f 1 ( t ) + f 2 ( t ) + · · · + f n ( t ) where each f i ( t ) is the product of a polynomial, an exponential function, and a trigonometric func- tion. That is, say that we can write f i ( t ) = d i X j =0 a ij t j e α i t cos β i t + d i X j =0 b ij t j e α i t sin β i t for some constants a ij , b ij , α i , and β i . Note that f i ( t ) involves a polynomial of degree d i . #2. Make a guess that a solution Y i = Y i ( t ) of the nonhomogeneous equation a ( t ) Y 00 i + b ( t ) Y 0 i + c ( t ) Y i = f i ( t ) for i = 1 , 2 , . . . , n ; is in the form Y i ( t ) = d i +2 X j =0 A ij t j e α i t cos β i t + d i +2 X j =0 B ij t j e α i t sin β i t for some constants A ij and B ij . Note that α i , and β i are the same as above, and that Y i ( t ) involves a polynomial of degree ( d i + 2). #3. Recombine as the sum Y ( t ) = Y 1 ( t ) + Y 2 ( t ) + · · · + Y n ( t ); then Y = Y ( t ) is the desired solution to the nonhomogeneous equation a Y 00 + b Y 0 + c Y = f ( t ) . This is known as the Method of Undetermined Coefficients .

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