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undetcoeff

# undetcoeff - 4 Substitute the trial solution into the...

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GUIDELINES FOR THE METHOD OF UNDETERMINED COEFFICIENTS Given the constant coefficient linear differential equation a ¨ x + b ˙ x + c x = f ( t ) , where f ( t ) is an exponential, a simple sinusoidal function, a polynomial, or a product of these functions: 1. Solve the homogeneous equation for a pair of linearly independent solutions x 1 ( t ) and x 2 ( t ). 2. If f ( t ) is not a solution of the homogeneous equation, take a trial solution of the same type as f ( t ) according to the suggestions given in class. 3. If f ( t ) is a solution of the homogeneous equation, take a trial solution of the same type as f ( t ) multiplied by the lowest power of t for which no term of the trial solution is a solution of the homogeneous equation.
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Unformatted text preview: 4. Substitute the trial solution into the diﬀerential equation and solve for the undetermined coeﬃcients so that it is a particular solution x p ( t ). 5. Set x ( t ) = x p ( t ) + c 1 x 1 ( t ) + c 2 x 2 ( t ), where the constants c 1 and c 2 can be determined if initial conditions are given. 6. If f ( t ) is a sum of forcing functions of the type described above, split the problem into simpler parts. Find a particular solution for each part, then add the particular solutions to obtain x p ( t ). 1...
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