Mathematical Methods (10/24.539)

II. Linear Differential Equations

Finding Particular Solutions

Undetermined Coefficients (UC)

The method of undetermined coefficients is rather straightforward but it is generally useful only for constant coefficient linear systems. In addition, the forcing function, r(x), must be a relatively simple function (exponential, polynomial, sinusoid, or a linear combination of such functions).

General rule: Choose yp(x) to have the same form as r(x) and all its derivatives. Evaluate the unknown coefficients within yp(x) by substitution into the original inhomogeneous ODE, and simply equate the coefficients of the like terms on both sides of the equations.

Special rule: If yp(x) via the general rule contains terms that are solutions to the homogeneous equations, one then multiplies this choice by x (from the Reduction of Order Method).

Example 2.1 provides an example of the method of Undetermined Coefficients.

Variation of Parameters (VOP)

A more general method for finding particular solutions is the variation of parameter technique. The method can be summarized as follows:

Given

with homogeneous solution

one can write the particular solution as

where

or, in general,

where the Wronskian for an nth order system is given by

and Wi(x) is the determinant obtained after replacing the ith column of the right hand side of W(x) with the vector . For example, in a 3rd order system, W2(x) is given by

The development of these equations for the specific case of a second order system is given in the next section. Generalization to higher order systems is straightforward.

Example 2.2 illustrates the Variation of Parameters method using the same problem statement as given in Example 2.1 (which used the Undetermined Coefficients approach).

Variation of Parameters - Detailed Development for 2nd Order Systems

This subsection derives the summary equations presented above for the Variation of Parameters method. The development here is specific to a second order system, but it is easily extended to higher order linear ODEs.

Let’s start with the general 2nd order equation written in standard form,

with homogeneous solution

Now assume yp of the form

which gives,

At this point we recognize that an addition constraint, beyond the original balance equation, will be needed because we are trying to find two unknowns, u1 and u2. For simplicity, let’s choose the following relationship,

which greatly simplifies the above equation for yp’, giving

Now the second derivative becomes

Substitution of these expressions into the original equation gives

Rearranging gives

Therefore, since both the first and second terms vanish (because y1 and y2 are solutions to the homogeneous equation), we have

Writing eqns. (2.59) and (2.62) in matrix form (two equations with two unknowns) gives

Using Cramer’s rule the solution to this system can be written as

Finally, integrating these latter expressions to give u1 and u2 allows one to write the desired expression for yp(x) as

This result is consistent with the general relationships given above.

Operator Notation

General Discussion

Linear ODEs are very common in engineering analysis. Because they occur so frequently, it is often convenient to use a shorthand notation to simplify the sometimes lengthy and very detailed differential equations that are characteristic of many real systems. In particular, let’s define the common derivative operation with the following notation:

The quantities are called differential operators and they have properties analogous to algebraic quantities. For example, a general 2nd order linear ODE can be written simply as Ly = r, where the L operator is the combination of three operators and is given by

This simplified notation is very convenient once one becomes comfortable with the interpretation of the operators utilized.

Two Points About Linear Operators

1. Linear operators with constant coefficients can be factored and the order of the operation is unimportant. For example, the following operator expressions are all equivalent:

2. Linear operators with variable coefficients can also be factored, but the order of operation is extremely important. To see this, consider the following two operator expressions that are nearly identical, except for the order of operation:

Case 1:

Case 2:

Thus, it is clear that Case 1 and Case 2 lead to different results. Therefore,

.

Summary Note

The point here is that linear operator notation can be very useful at times, and it is used extensively in the literature -- so the student must become familiar with its use. In addition, proper factorization of linear operators can also help in the solution of some very complicated systems. Example 2.3 further explains this approach. However, for variable coefficient systems, the order of operation is very important and care must be exercised when working with such systems!

10/24.539 Lecture Notes by Dr. J. R. White, UMass-Lowell (updated August 1998).

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