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The Method of Undetermined Coefficients

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## The Method of Undetermined Coefficients

❶All that we need to do it go back to the appropriate examples above and get the particular solution from that example and add them all together. This work is avoidable if we first find the complementary solution and comparing our guess to the complementary solution and seeing if any portion of your guess shows up in the complementary solution.
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In this section we introduce the method of undetermined coefficients to find particular solutions to nonhomogeneous differential equation. We work a wide variety of examples illustrating the many guidelines for making the initial guess of the form of the particular solution that is needed for the method.

Pros and Cons of the Method of Undetermined Coefficients:The method is very easy to perform. However, the limitation of the method of undetermined coefficients is that the non-homogeneous term can only contain simple functions such as,,, and so the trial function can be effectively guessed.

The Method of Undetermined Coefficients In order to give the complete solution of a nonhomogeneous linear differential equation, Theorem B says that a particular solution must be added to the general solution of the corresponding homogeneous equation. For complex equations, the annihilator method or variation of parameters is less time consuming to perform. Undetermined coefficients is not as general a method as variation of parameters, since it only works for differential equations that follow certain forms.

GUIDELINES FOR THE METHOD OF UNDETERMINED COEFFICIENTS Given a constant coe cient linear di erential equation ay00+ by0+ cy = g(t); where gis 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: fy 1;y 2g. 2. The method of undetermined coefficients is a technique for determining the particular solution to linear constant-coefficient differential equations for certain types of nonhomogeneous terms f(t).