Same Roots In A Differential Equations

In the realm of differential equations, the concept of "same roots" – often referred to as repeated roots or multiple roots – plays a pivotal role in determining the nature and form of solutions. When dealing with linear homogeneous differential equations, particularly those with constant coefficients, the roots of the characteristic equation dictate the general solution. The occurrence of repeated roots introduces complexities and necessitates special techniques to derive the complete solution set. Understanding the implications of repeated roots is crucial for solving a wide array of problems in physics, engineering, and applied mathematics, where differential equations serve as fundamental models.

Understanding Differential Equations and Characteristic Equations

Before diving into the specifics of repeated roots, it's essential to establish a firm understanding of differential equations and characteristic equations.

What is a Differential Equation?

A differential equation is a mathematical equation that relates a function with its derivatives. In simpler terms, it's an equation that involves an unknown function and its rates of change. Differential equations are broadly classified into two main types: ordinary differential equations (ODEs) and partial differential equations (PDEs).

Ordinary Differential Equations (ODEs): These equations involve functions of only one independent variable and their derivatives. For example:
```
dy/dx + 2y = x
```
Partial Differential Equations (PDEs): These equations involve functions of multiple independent variables and their partial derivatives. For example:
```
∂u/∂t = α(∂²u/∂x²)
```
This article will focus on ODEs, specifically linear homogeneous ODEs with constant coefficients.

Linear Homogeneous ODEs with Constant Coefficients

A linear homogeneous ODE with constant coefficients has the general form:

aₙy⁽ⁿ⁾ + aₙ₋₁y⁽ⁿ⁻¹⁾ + ... + a₁y' + a₀y = 0

Where:

y is the dependent variable (the function we're trying to find).
x is the independent variable.
y⁽ⁿ⁾ denotes the nth derivative of y with respect to x.
aₙ, aₙ₋₁, ..., a₁, a₀ are constant coefficients.

For example, the following is a linear homogeneous ODE with constant coefficients:

2y'' + 5y' - 3y = 0

The Characteristic Equation

To solve a linear homogeneous ODE with constant coefficients, we often employ the concept of a characteristic equation. This equation is derived by assuming a solution of the form:

y = e^(rx)

Where r is a constant to be determined. Substituting this assumed solution into the ODE and simplifying, we obtain a polynomial equation in terms of r, known as the characteristic equation or auxiliary equation.

For the general form of the ODE mentioned above, the characteristic equation is:

aₙrⁿ + aₙ₋₁rⁿ⁻¹ + ... + a₁r + a₀ = 0

The roots of this characteristic equation, denoted as r₁, r₂, ..., rₙ, determine the form of the general solution to the ODE.

Repeated Roots: When the Characteristic Equation Gets Tricky

When solving differential equations using the characteristic equation method, we typically encounter three scenarios regarding the roots:

Distinct Real Roots: Each root is a different real number.
Complex Conjugate Roots: Roots appear in pairs of the form a + bi and a - bi, where 'a' and 'b' are real numbers and 'i' is the imaginary unit (√-1).
Repeated Roots: At least one root appears more than once. This is the main focus of this article.

Repeated roots occur when the characteristic equation has factors that are raised to a power greater than one. For instance, consider the following characteristic equation:

(r - 2)²(r + 1) = 0

This equation has roots r = 2 (with multiplicity 2) and r = -1 (with multiplicity 1). The root r = 2 is a repeated root.

Deriving the General Solution with Repeated Roots

The presence of repeated roots requires a modification of the standard approach for constructing the general solution. If we have a root r that is repeated k times, we cannot simply use e^(rx) as k independent solutions. This is because these solutions would be linearly dependent.

The Standard Approach Fails

If we naively apply the distinct roots method to repeated roots, we end up with linearly dependent solutions. For example, if r = 2 is a repeated root with multiplicity 2, we might think that e^(2x) and e^(2x) are two independent solutions, but they are clearly the same function (multiplied by a constant).

The Correct Approach: Multiplying by Powers of x

To obtain linearly independent solutions in the case of repeated roots, we multiply e^(rx) by increasing powers of x up to x^(k-1), where k is the multiplicity of the root.

If r is a repeated root with multiplicity k, then the k linearly independent solutions are:

e^(rx), xe^(rx), x²e^(rx), ..., x^(k-1)e^(rx)

The general solution is then a linear combination of these solutions, along with the solutions corresponding to any other distinct roots.

Examples of Solving Differential Equations with Repeated Roots

Let's illustrate this with a few examples:

Example 1: Second-Order ODE

Consider the following second-order linear homogeneous ODE with constant coefficients:

y'' - 4y' + 4y = 0

Form the Characteristic Equation:

The characteristic equation is:
```
r² - 4r + 4 = 0
```
Solve the Characteristic Equation:

This equation can be factored as:
```
(r - 2)² = 0
```
Thus, we have a repeated root r = 2 with multiplicity k = 2.
Construct the General Solution:

Since we have a repeated root with multiplicity 2, the two linearly independent solutions are:
```
e^(2x) and xe^(2x)
```
Therefore, the general solution is:
```
y(x) = c₁e^(2x) + c₂xe^(2x)
```
Where c₁ and c₂ are arbitrary constants determined by initial conditions (if provided).

Example 2: Third-Order ODE

Consider the following third-order linear homogeneous ODE with constant coefficients:

y''' - 5y'' + 8y' - 4y = 0

Form the Characteristic Equation:

The characteristic equation is:
```
r³ - 5r² + 8r - 4 = 0
```
Solve the Characteristic Equation:

This equation can be factored as:
```
(r - 1)(r - 2)² = 0
```
Thus, we have roots r = 1 (with multiplicity 1) and r = 2 (with multiplicity 2).
Construct the General Solution:

For the root r = 1, we have the solution e^(x). For the root r = 2 (with multiplicity 2), we have the solutions e^(2x) and xe^(2x).

Therefore, the general solution is:
```
y(x) = c₁e^(x) + c₂e^(2x) + c₃xe^(2x)
```
Where c₁, c₂, and c₃ are arbitrary constants.

Example 3: Fourth-Order ODE

Consider the following fourth-order linear homogeneous ODE with constant coefficients:

y'''' - 2y''' + y'' = 0

Form the Characteristic Equation:

The characteristic equation is:
```
r⁴ - 2r³ + r² = 0
```
Solve the Characteristic Equation:

This equation can be factored as:
```
r²(r - 1)² = 0
```
Thus, we have roots r = 0 (with multiplicity 2) and r = 1 (with multiplicity 2).
Construct the General Solution:

For the root r = 0 (with multiplicity 2), we have the solutions e^(0x) = 1 and xe^(0x) = x. For the root r = 1 (with multiplicity 2), we have the solutions e^(x) and xe^(x).

Therefore, the general solution is:
```
y(x) = c₁ + c₂x + c₃e^(x) + c₄xe^(x)
```
Where c₁, c₂, c₃, and c₄ are arbitrary constants.

Why Does This Method Work? A Glimpse into the Theory

The reason why multiplying by powers of x yields linearly independent solutions can be understood through a deeper dive into the theory of linear differential equations.

Linear Independence and the Wronskian

The concept of linear independence is fundamental. A set of functions f₁, f₂, ..., fₙ is linearly independent if the only solution to the equation:

c₁f₁(x) + c₂f₂(x) + ... + cₙfₙ(x) = 0

is c₁ = c₂ = ... = cₙ = 0 for all x in the interval of interest.

The Wronskian is a determinant that can be used to test for linear independence of a set of solutions to a differential equation. For n functions f₁, f₂, ..., fₙ, the Wronskian is defined as:

W(f₁, f₂, ..., fₙ)(x) = det
\begin{bmatrix}
f₁(x) & f₂(x) & ... & fₙ(x) \\
f₁'(x) & f₂'(x) & ... & fₙ'(x) \\
... & ... & ... & ... \\
f₁⁽ⁿ⁻¹⁾(x) & f₂⁽ⁿ⁻¹⁾(x) & ... & fₙ⁽ⁿ⁻¹⁾(x)
\end{bmatrix}

If the Wronskian is non-zero for at least one point in the interval of interest, then the functions are linearly independent.

Applying the Wronskian to Repeated Roots

When we consider solutions of the form e^(rx), xe^(rx), x²e^(rx), ... , x^(k-1)e^(rx) derived from a repeated root r, calculating the Wronskian will show that it is indeed non-zero. This confirms that these solutions are linearly independent and can form the basis for the general solution.

The Connection to Reduction of Order

The method of multiplying by powers of x is closely related to a technique called reduction of order. If we know one solution y₁(x) to a linear homogeneous ODE, reduction of order provides a way to find a second linearly independent solution y₂(x) by assuming y₂(x) = v(x)y₁(x), where v(x) is a function to be determined. When applied to the case of repeated roots, this method leads to solutions of the form xe^(rx), x²e^(rx), and so on.

Practical Applications and Significance

Understanding and correctly handling repeated roots in differential equations is crucial in various fields:

Physics: In mechanics, damped harmonic oscillators with critical damping exhibit repeated roots in their characteristic equation. The solution describes the system returning to equilibrium as quickly as possible without oscillating.
Engineering: In electrical circuits, analyzing the behavior of circuits with specific component values can lead to differential equations with repeated roots. This helps engineers design circuits with desired response characteristics.
Control Systems: In control theory, repeated roots appear in the analysis of system stability and response. Understanding their impact allows for the design of stable and well-behaved control systems.
Mathematical Modeling: Many real-world phenomena can be modeled using differential equations. Accurately capturing the dynamics of these phenomena often requires solving equations with repeated roots.

Common Mistakes and How to Avoid Them

Working with repeated roots can be tricky, and several common mistakes can lead to incorrect solutions:

Failing to Recognize Repeated Roots: Always carefully solve the characteristic equation to identify all roots and their multiplicities.
Using Linearly Dependent Solutions: Do not simply use e^(rx) multiple times for a repeated root. Remember to multiply by increasing powers of x.
Incorrectly Applying Initial Conditions: After finding the general solution, carefully apply the initial conditions to solve for the arbitrary constants.
Algebraic Errors: Double-check your algebraic manipulations, especially when solving the characteristic equation and applying the product rule when differentiating xe^(rx), x²e^(rx), etc.

Conclusion

Repeated roots in the characteristic equation of a linear homogeneous differential equation require a specific approach to find the general solution. By multiplying the exponential solution e^(rx) by increasing powers of x up to x^(k-1), where k is the multiplicity of the root, we can obtain a set of k linearly independent solutions. Understanding the underlying theory and practicing with examples is essential for mastering this technique and applying it effectively to solve a wide range of problems in science and engineering. Recognizing the significance of repeated roots allows for accurate modeling and analysis of systems exhibiting critical damping, resonance, and other complex behaviors. The ability to confidently handle repeated roots is a valuable asset in the toolkit of anyone working with differential equations.