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Particular Solutions of Nonhomogeneous D.E.’s (continued)

For simple functions, f(x) one can find particular solutions y_p of nonhomogeneous,

linear differential equations with constant coefficients using the method of

undetermined coefficients. Consider the following d.e.

y ’’ + A y ’ + B y = f(x)

First Step: Take the derivative of f(x) up to the highest order in the d.e. In the general

case above it is a second order d.e. Then the particular solution is a linear combination

of the different functions obtained from the right hand side of the d.e.. i.e. If the right

hand side happened to be 23 sin 2x, then the assumed form of the particular solution

would be y_p = A sin 2x + B cos 2x since both sin 2x and cos 2x appear when

taking the function, f(x), and its first and second derivatives. The assumption here is

that the complementary solution to the homogeneous d.e. does not contain either

sin 2x nor cos 2x.

The cases below summarize the simple cases for f(x) for which one can apply the

method of undetermined coefficients to find the particular solution of nonhomogeneous

d.e.’s.

Case 1: f(x) = polynomial = a₁x ^m + a₂x ^{m -1} + . . . . + a_n

y_p(x) =A_mx ^m + A_m-1x ^{m
-1} + . . . . + A_o

Case 2: f(x) = a cos kx + b sin kx

y_p(x) = A cos kx + B sin kx

Case 3: f(x) = a e ^kx + b e ^–kx

y_p (x) = A e ^kx + B e ^–kx

Case 4: Note - f(x) could involve products of the functions in Cases 1 – 3

NOTE: If any of the f(x) appear in the complementary solution, then you need to pick

a new function in your particular solution that is linearly independent. i.e.

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