Characteristic equation and general solution of constant-coefficient linear differential equations

Introduction

Constant-coefficient linear differential equations are fundamental in mathematics and its applications, particularly in physics, engineering, and control theory. These equations take the general form:
$$ a_n y^{(n)} + a_{n-1} y^{(n-1)} + \cdots + a_1 y' + a_0 y = 0,
$$
where $ a_n, a_{n-1}, \dots, a_0 $ are constants and $ y(t) $ is the unknown function. The structure of these equations ensures that their solutions can be analyzed using the concept of characteristic equations. The characteristic equation is derived by assuming solutions of the form $ y(t) = e^{rt} $, where $ r $ is a constant to be determined. This substitution simplifies the differential equation into a polynomial equation in $ r $, enabling the determination of the system's behavior.

The characteristic equation is pivotal because it encapsulates the essential information about the differential equation's roots, which dictate the form of the general solution. By analyzing the roots, we can classify the nature of the solutions and predict their stability, oscillatory behavior, or exponential decay. This article explores the derivation of the characteristic equation, its roots, and the implications of these roots on the general solution of constant-coefficient linear differential equations.

Derivation of the Characteristic Equation

To derive the characteristic equation, we assume a solution of the form $ y(t) = e^{rt} $, where $ r $ is a constant. Substituting this into the differential equation yields:
$$ a_n r^n e^{rt} + a_{n-1} r^{n-1} e^{rt} + \cdots + a_1 r e^{rt} + a_0 e^{rt} = 0.
$$
Dividing through by $ e^{rt} $, the equation reduces to:
$$ a_n r^n + a_{n-1} r^{n-1} + \cdots + a_1 r + a_0 = 0.
$$
This polynomial equation in $ r $ is known as the characteristic equation. The roots of this equation determine the form of the general solution. For example, if the roots are real and distinct, the general solution is a linear combination of exponential functions. If the roots are complex conjugates, the solution involves sinusoidal functions. Repeated roots require special treatment, as they lead to solutions with polynomial factors.

Analysis of Roots and General Solution

The roots of the characteristic equation are critical in determining the nature of the differential equation's solution. The analysis of these roots can be divided into three primary cases:

1. Distinct Real Roots

If the characteristic equation has distinct real roots $ r_1, r_2, \dots, r_k $, the general solution of the differential equation is given by:
$$ y(t) = C_1 e^{r_1 t} + C_2 e^{r_2 t} + \cdots + C_k e^{r_k t},
$$
where $ C_1, C_2, \dots, C_k $ are constants of integration. The exponential functions $ e^{rt} $ decay or grow depending on the sign of $ r $, and the constants $ C_i $ determine the amplitude of each term. This solution is straightforward and captures the behavior of the system in terms of exponential growth or decay.

2. Repeated Real Roots

When the characteristic equation has a repeated real root $ r $, the general solution includes an additional term involving $ t $ to account for the multiplicity of the root. The solution takes the form:
$$ y(t) = (C_1 + C_2 t) e^{rt}.
$$
The presence of the polynomial $ t $ reflects the system's response to repeated roots, which can lead to non-oscillatory behavior. This case is essential in understanding systems with damping or resonance, where the solution may exhibit slower decay or sustained oscillations.

3. Complex Conjugate Roots

If the characteristic equation has complex conjugate roots $ \alpha \pm \beta i $, where $ \alpha $ and $ \beta $ are real numbers, the general solution involves trigonometric functions. The solution is expressed as:
$$ y(t) = e^{\alpha t} \left[ C_1 \cos(\beta t) + C_2 \sin(\beta t) \right].
$$
The exponential $ e^{\alpha t} $ determines the system's transient behavior, while the cosine and sine terms describe the steady-state oscillations. The real part $ \alpha $ dictates