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Undetermined Coefficients: Method & Examples

Instructor: Norair Sarkissian
The method of undetermined coefficients is used to solve a class of nonhomogeneous second order differential equations. This method makes use of the characteristic equation of the corresponding homogeneous differential equation.

Introduction

Differential equations are mathematical equations which represent a relationship between a function and one or more of its derivatives. Differential equations are used to mathematically model economics, physics and engineering problems. Examples include mechanics, where we use such equations to model the speed of moving objects (such as cars or projectiles), as well as electronics, where differential equations are employed to relate voltages and currents in a circuit.

While calculus offers us many methods for solving differential equations, there are other methods that transform the differential equation, which is a calculus problem, into an algebraic equation. The Laplace transform method is just such a method, and so is the method examined in this lesson, called the method of undetermined coefficients.

The term 'undetermined coefficients' is based on the fact that the solution obtained will contain one or more coefficients whose values we do not generally know. In addition to the coefficients whose values are not determined, the solution found using this method will contain a function which satisfies the given differential equation. For instance, let's say that in the process of solving a differential equation, we obtain a solution containing the undetermined coefficients A, B and C, given by

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This means that for any values of A, B and C, the function y(t) satisfies the differential equation. For example, we could set A = 1, B = 1 and C=2, and discover that the solution,

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satisfies the differential equation. It also means that any other set of values for these constants, such as A = 2, B = 3 and C = 1, or A = 1, B = 0 and C = 17, would also yield a solution. Therefore, the following functions are solutions as well:

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Thus, we can see that by making use of undetermined coefficients, we are able to find a family of functions which all satisfy the differential equation, no matter what the values of these unknown coefficients are.

To learn more about the method of undetermined coefficients, we need to make sure that we know what second order homogeneous and nonhomogeneous equations are.

Review: Homogeneous Equations

A homogeneous second order differential equation is of the form

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where a, b and c are constants.

The solution of such an equation involves the characteristic (or auxiliary) equation of the form

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Depending on the sign of the discriminant of the characteristic equation, the solution of the homogeneous differential equation is in one of the following forms:

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Nonhomogeneous Equations

But is it possible to solve a second order differential equation when the right-hand side does not equal zero? An equation of the form

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where g(t) is nonzero, is called a nonhomogeneous equation.

It turns out that if the function g(t) on the right hand side of the nonhomogeneous differential equation is of a special type, there is a very useful technique known as the method of undetermined coefficients which provides us with a unique solution that satisfies the differential equation. This unique solution is called the particular solution of the equation.

The Method Of Undetermined Coefficients

Let us consider the special case whereby the right-hand side of the nonhomogeneous differential equation is of the form

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Here n is a nonnegative integer (i.e., n can be either positive or zero), r is any real number, and C is a nonzero real number.

Let's consider the following example:

If C = 6, n = 2 and r = 4, the right-hand side of the equation equals

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The method of undetermined coefficients can be applied when the right-hand side of the differential equation satisfies this form. It provides us with a particular solution to the equation.

The method of undetermined coefficients states that the particular solution will be of the form

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Please note that this solution contains at least one constant (in fact, the number of constants is n+1):

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The exponent s is also a constant and takes on one of three possible values: 0, 1 or 2. Its value represents the number of matches between r and the roots of the characteristic equation. To be more specific, the value of s is determined based on the following three cases.

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Thus, if r is not a solution of the characteristic equation (so there is no match), then we set s = 0.

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So, if r is a simple (or single) root of the characteristic equation (we have a single match), then we set s = 1.

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