Characteristic equation (calculus)

The characteristic equation in the theory of differential equations a means to calculate solutions of linear differential equations with constant coefficients. By the determination of a fundamental system of the differential equation is reduced to the solution of a polynomial equation.

An analogous method can also be used to solve linear difference equations with constant coefficients.

Leonhard Euler reported this method for solving differential equations in the case of constant coefficients in 1739 in a letter to Johann Bernoulli, even without considering multiple solutions of the characteristic equation. A solution to a differential equation with multiple roots in the characteristic equation is found but then later in Euler's Institutiones calculi integralis. Augustin- Louis Cauchy Next Gaspard Monge and have researched it.

Definition

Given a homogeneous linear differential equation of order of the form

For an unknown function with constant complex coefficients. Then is the associated characteristic equation

The polynomial

On the left side of the equation and the characteristic polynomial of the differential equation is known.

Thus, formally, the characteristic equation is obtained by replacing the -th derivative of the -th power of the polynomial variable ( referred to herein ).

Solutions

According to the theory of linear differential equations, the solution set of a homogeneous linear differential equation of order forms one -dimensional vector space. Thus, it suffices to find linearly independent solutions of the differential equation for the determination of the general solution. By the fundamental theorem of algebra, the polynomial has exactly complex roots, if you count them according to their multiplicity. Below is shown how using these solutions of the characteristic equation always a basis of the solution space of the differential equation, ie, a fundamental system, can be specified.

Simple Solutions

The approach leads to an unknown due to the equation and thus after division by the characteristic structure equation. Thus:

Now, if all the roots are distinct, one gets in this way, different solutions of the differential equation and it can be shown that they are also linearly independent. The general solution is therefore in this case

With arbitrary constants.

Multiple solutions

However, if a multiple solution of the characteristic equation, we obtain in this way is only one solution, so no more fundamental system. In this case, however, more linearly independent solutions can be specified in a simple manner:

Complex solutions for real equation

In the following, all the coefficients are real numbers. In this case one is often interested only in the real solution of the equation, and thus also in a real fundamental system. It has a complex solution, then the complex conjugate is also a solution. These correspond to linearly independent complex solutions of the differential equation. With the help of Euler's formula is obtained from this

And

As real solutions to the differential equation. These are also linearly independent. Analogously, in the case of multiple complex solutions respectively by the transition to the real and imaginary parts for each pair of complex conjugate solutions construct two linearly independent real solutions. To emerge from the complex conjugate solutions the two real solutions.

Examples

• The characteristic equation of the differential equation is and has the solutions and. This yields the fundamental system, and the general solution of the differential equation is.
• The vibration equation has the characteristic equation with the complex conjugate solutions. A complex fundamental system, then, is a real.
• The differential equation