Math Help

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

In a Nut Shell: For functions, f(x), involving polynomials, trig functions, exponential functions,

and products of these functions, you can find particular solutions, y_p , of nonhomogeneous,

linear differential equations with constant coefficients. Quotients are not included.

y ’’ + A y ’ + B y = f(x) using the method of undetermined coefficients.

Strategy: Take the derivative of f(x) up to the highest order of the d.e. For the second

order d.e. illustrated above, you need to take two derivatives of f(x). Then the particular

solution is a linear combination of the different functions obtained from these derivatives.

For example if f(x) 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 first and second derivatives of 23 sin 2x under the assumption that the complementary

solution to the homogeneous d.e. does not contain either sin 2x nor cos 2x.

The table below summarize 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) in the particular solution appear in the complementary solution,
             then you need to create new functions in your particular solution that are linearly
              independent of those in the homogeneous solution by multiplying f(x) by an
              appropriate value of x such as x, x², x³, etc. to obtain linearly independent functions.

Click here for examples using the method of undetermined coefficients.

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