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Unformatted text preview: GUIDELINES FOR THE METHOD OF UNDETERMINED COEFFICIENTS Given a constant coefficient linear differential equation ay 00 + by + cy = g ( x ) , where g is an exponential, a simple sinusoidal function, a polynomial, or a product of these functions: 1. Find a pair of linearly independent solutions of the homogeneous problem: { y 1 , y 2 } . 2. If g is NOT a solution of the homogeneous equation, take a trial solution of the same type as g as suggested in the table below: Forcing Function Trial Solution ae rt Ae rt a sin ( ωt ) or a cos ( ωt ) A sin ( ωt ) + B cos ( ωt ) at n P ( t ) n a positive integer P a general polynomial of degree n at n e rt P ( t ) e rt n a positive integer P a general polynomial of degree n t n [ a sin ( ωt ) + b cos ( ωt )] P ( t )[ A sin ( ωt ) + B cos ( ωt )] n a positive integer P a general polynomial of degree n e rt [ a sin ( ωt ) + b cos ( ωt )] e rt [ A sin ( ωt ) + B cos ( ωt )] 3. If g is a solution of the homogeneous problem, take a trial solution of the same type as g multiplied by the lowest power of t for which NO TERM of the trial solution is a solution of the homogeneous equation. 4. Substitute the trial solution into the differential equation and solve for the undetermined coefficients so that it is a particular solution y p . 5. Set y ( t ) = y p ( t ) + [ c 1 y 1 ( t ) + c 2 y 2 ( t )] where the constants c 1 and c 2 can be determined if initial conditions are given. 6. If g is a sum of the type of forcing function described above, split the problem into simpler parts. Find a particular solution for each of these, then add particular solutions to obtain y p for the complete equation....
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This note was uploaded on 12/02/2009 for the course MATH 352 taught by Professor Staff during the Spring '08 term at University of Delaware.
 Spring '08
 Staff

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