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Unformatted text preview: Math 334 Lecture #16 § 3.5,4.3: Method of Undetermined Coefficients This is also known as the “method of guessing” as it applies to n th-order linear nonho- mogeneous ODE’s with constant coefficients: L [ y ] = a n y ( n ) + a n- 1 y ( n- 1) + · · · + a 1 y + a y = g ( t ) . The differential operator L takes in a function y and puts out the function L [ y ]. What form does Y p have to have so that L [ Y p ] would be anything like g ( t )? Basic Rule . Let L [ y ] = y 00 + 4 y and g ( t ) = cos t . If A is a constant, then L [ A cos t ] =- A cos t + 4 A cos t = 3 A cos t. What would be a form for a particular solution, Y p ? It would be Y p = A cos t . What value of A makes Y p a particular solution of L [ y ] = g ( t )? It is A = 1 / 3. Check this choice of Y p : L [ Y p ] =- (1 / 3) cos t + (4 / 3) cos t = cos t = g ( t ) X . Modification Rule . Let L [ y ] = y 00- 3 y + 2 y and g ( t ) = e t ....
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This note was uploaded on 03/08/2011 for the course MATH 334 taught by Professor Smith during the Spring '11 term at Vanderbilt.
- Spring '11