lecture_19 - MA 36600 LECTURE NOTES: MONDAY, MARCH 2 Method...

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Unformatted text preview: 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 ay 00 + by + cy = 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 i t cos i t + d i X j =0 b ij t j e i t sin i t for some constants a ij , b ij , i , and 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 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 i t cos i t + d i +2 X j =0 B ij t j e i t sin i t for some constants A ij and B ij . Note that i , and 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 aY 00 + bY + cY = f ( t ) ....
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This note was uploaded on 05/24/2011 for the course MA 36600 taught by Professor Ma during the Spring '09 term at Purdue University-West Lafayette.

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lecture_19 - MA 36600 LECTURE NOTES: MONDAY, MARCH 2 Method...

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