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.
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
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,
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:
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
where a, b and c are constants.
The solution of such an equation involves the characteristic (or auxiliary) equation of the form
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:
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
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
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
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
Please note that this solution contains at least one constant (in fact, the number of constants is n+1):
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.
Thus, if r is not a solution of the characteristic equation (so there is no match), then we set s = 0.
So, if r is a simple (or single) root of the characteristic equation (we have a single match), then we set s = 1.
This is the case where r is a double root of the characteristic equation, i.e., we have a double match; hence, we set s = 2.
Find a particular solution to the differential equation
We first check to see whether the right hand side of the differential equation is of the form for this method to be applied.
By comparing both sides of the equation, we can see that they are equal when
We now consider the homogeneous form of the given differential equation; i.e., we temporarily set the right-hand side of the equation to zero. This gives us the homogeneous equation
whose characteristic equation is
We can find the roots of this equation using factoring, as the left hand side of this equation can be factored to yield the equation
Therefore, the two distinct roots of the characteristic equation are
In step 3 below, we will use these solutions to determine the value of the exponent s in the particular solution.
We now return to the nonhomogeneous equation. We note that we have
Therefore, r is a simple root of the characteristic equation, we apply case (2) and set s = 1.
The particular solution of this non-homogeneous equation is
We have already determined that
Since n = 0, the expression in parentheses consists of just one constant, namely:
Therefore, the particular solution of the differential equation is
We have discovered that a special category of second order nonhomogeneous differential equations can be solved using the method of undetermined coefficients. This method allows us to find a particular solution to the differential equation. It requires the solution of the corresponding homogeneous equation, including the generation of the characteristic equation. While this method cannot be used to solve all nonhomogeneous second order equations, it does provide us with a particular solution whenever the right hand side of the equation is of the form: