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7. UNDETERMINED COEFFICIENTS

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7. UNDETERMINED COEFFICIENTS

As was discussed In Section 5, the general solution of (1) is 𝑦 = 𝑦𝑐 + 𝑦𝑝. The complementary function 𝑦𝑐 is the general solution of the associated homogeneous DE of (1), that is,

In Section 6, we saw how to solve these kinds of equations when the coefficients were constants. Our goal in the present section is to develop a method for obtaining particular

(2)

Definition

Now, let us consider the following nonhomogeneous differential equation with constant coefficients:

(3)
(4)
(5)
(6)
(7)
(8)
(9)

The Method Step by Step

› Step 1: Solve the homogeneous equation and write the set of

independent

solution’s set, i.e. Fundamental Set of Solutions (FSS).

› Step 2: Find UC set (𝑆) of the right hand side function (𝐹(𝑥)).

› Step 3: If the set 𝑆 includes one or more member of FSS, then

multiply each member of

𝑆 by the lowest positive integer power of 𝑥.

So, the new set

, 𝑆

1

, does not include any member of FSS.

› Step 4: The linear combination of 𝑆

1

, is in the form of particular

solution

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CAUTION

Keep in mind that the method of undetermined coefficients

applies only to nonhomogeneities that are polynomials,

exponentials, sines or cosines, or products of these

functions.

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