A characteristic equations is an equation of degree upon which depends on the solution of a given th-order differential equations or difference equations. The characteristic equations can only formed when the differential or difference equation is homogeneous, linear, and has constant coefficients. 1

For a differential equation We will have a characteristic equation of the form

whose solutions are the roots from which the general solution can be formed. 1

Derivation

Starting with a linear homogeneous differential equation with constant coefficients We can see that is a solution because the exponential function has the property that its derivative has the same shape as the original function (see Eigenfunction). In order to solve for , one can substitute and its derivatives into the differential equation to get

Since can never be zero, we can divide it out, and we get the characteristic equation 1

Formation of the General Solution

Solving the characteristic equation allow one to find the general solution of the differential equation. The solution depends on where the roots are at the complex frequency plane, and whether there are repeated roots.

Distinct Real Root

The superposition principle of a linear system says that if are linearly independent solutions to a linear differential equation, then is also a solution for all constant values . Thus, if the characteristic equation has distinct real roots , then a general solution will be of the form

1

Repeated Real Root

If the characteristic equation has a root that repeated times, the general solution that corresponding to is

Example

thus

Complex Root

What if a root is complex? If a second-order differential equation has a characteristic equation with complex conjugate roots and , then the general solution is the following:

We can simplify it further with Euler’s formula:

Since and are arbitrary constants that may be complex, we can substitute and with some other constants. As a result,

Note that both and may be complex numbers.

Real Solutions

Sometimes we want to only find real solutions, in this case we can simplify the formula a bit further.

will only be real when and are complex conjugate (so will be a pure imaginary and will be real). In that case, let

This yields

2

Relation with Eigenvectors and Eigenvalues

In linear algebra, we also have the characteristic equation in the form of

where and are called eigenvalues and eigenvectors of the matrix , respectively. This concept is closely linked to characteristic equations in differential equations.

For a linear, homogeneous differential equation with constant coefficients:

The characteristic equation is

We can turn this nth order differential equation into a first-order system of differential equation by setting .

This gives us: and .

For example. for the equation , we get the following system:

The characteristic polynomial of this matrix is:

We get back the characteristic equation of the original differential equation (up to a multiplication of -1), demonstrating the direct link between the two concepts. 3

Footnotes

  1. Characteristic equation (calculus) - Wikipedia 2 3 4

    • Linear Systems and Signals, 3rd Edition, 2.2
  2. What is the relation between the characteristic equation in linear algebra and the characteristic equation discussed in differential equations? - Quora