4.3 - Math 334 Lecture#16 3.5,4.3 Method of Undetermined...

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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 0 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 + 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 ) . Modification Rule . Let L [ y ] = y - 3 y + 2 y and g ( t ) = e t . By the basic rule, the form of Y p should be Y p = Ae t . An equation that is satisfied by the undetermined coefficient A is L [ Y p ] = e t Ae t - 3 Ae t + 2 Ae t = e t 0 = e t .
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