Linearity And Order Of The Differential Equation T⁵y' - T³y'' + 6y = 0

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Determine if the equation t⁵y' - t³y'' + 6y = 0 is linear or non-linear and what its order is.

Understanding Differential Equations: A Comprehensive Analysis

In the realm of mathematics, differential equations play a pivotal role in modeling various phenomena across diverse fields, including physics, engineering, economics, and biology. These equations describe the relationship between a function and its derivatives, providing a powerful tool for understanding dynamic systems. In this article, we will delve into the intricacies of a specific differential equation, t⁵y' - t³y'' + 6y = 0, to determine its linearity and order. This exploration will not only enhance our understanding of this particular equation but also provide a framework for analyzing other differential equations.

To effectively analyze differential equations, it is crucial to grasp the fundamental concepts of linearity and order. These properties dictate the behavior and complexity of the equation, guiding the selection of appropriate solution techniques. Let's begin by dissecting these key concepts.

Linearity: A Cornerstone of Differential Equation Analysis

The linearity of a differential equation is a critical property that determines the applicability of various solution methods. A differential equation is deemed linear if it satisfies two fundamental principles:

  1. Additivity: If y₁(t) and y₂(t) are solutions to the differential equation, then their sum, y₁(t) + y₂(t), is also a solution.
  2. Homogeneity: If y(t) is a solution to the differential equation, then any constant multiple of y(t), denoted as cy(t) where c is a constant, is also a solution.

In simpler terms, a linear differential equation can be expressed in the following general form:

aₙ(t)y⁽ⁿ⁾(t) + aₙ₋₁(t)y⁽ⁿ⁻¹⁾(t) + ... + a₁(t)y'(t) + a₀(t)y(t) = f(t)

where:

  • y⁽ⁿ⁾(t) represents the nth derivative of the function y(t) with respect to t.
  • aₙ(t), aₙ₋₁(t), ..., a₁(t), a₀(t) are coefficient functions that depend only on the independent variable t.
  • f(t) is a function of the independent variable t, often referred to as the forcing function.

Conversely, a differential equation is classified as nonlinear if it fails to adhere to either the additivity or homogeneity principle. Nonlinearities can arise from various sources, including:

  • Nonlinear terms involving the dependent variable y(t) or its derivatives, such as y(t)², sin(y(t)), or (y'(t))³.
  • Products of the dependent variable y(t) and its derivatives, such as y(t)y'(t).
  • Nonlinear functions of the dependent variable y(t) or its derivatives, such as e^(y(t)) or ln(y'(t)).

The presence of any of these nonlinearities renders the differential equation nonlinear, often making it significantly more challenging to solve analytically.

Order: Quantifying the Complexity of a Differential Equation

The order of a differential equation is another crucial characteristic that dictates its complexity and the number of arbitrary constants required in its general solution. The order is simply defined as the highest order derivative present in the equation. For instance:

  • A first-order differential equation involves only the first derivative, y'(t).
  • A second-order differential equation involves the second derivative, y''(t), as the highest order derivative.
  • An nth-order differential equation involves the nth derivative, y⁽ⁿ⁾(t), as the highest order derivative.

The order of a differential equation directly influences the number of arbitrary constants that appear in its general solution. Specifically, the general solution of an nth-order differential equation will contain n arbitrary constants. These constants arise from the process of integration, where each integration introduces an arbitrary constant.

Understanding the order of a differential equation is essential for determining the appropriate solution techniques and interpreting the general solution.

Analyzing the Given Differential Equation: t⁵y' - t³y'' + 6y = 0

Now, let's apply our understanding of linearity and order to the specific differential equation at hand: t⁵y' - t³y'' + 6y = 0.

Determining Linearity

To ascertain the linearity of the given equation, we must examine whether it satisfies the additivity and homogeneity principles. Let's analyze each principle in detail.

Additivity

Suppose y₁(t) and y₂(t) are two solutions to the differential equation. This means they satisfy the equation when substituted for y(t):

t⁵y₁'(t) - t³y₁''(t) + 6y₁(t) = 0
t⁵y₂'(t) - t³y₂''(t) + 6y₂(t) = 0

Now, let's consider the sum of these solutions, y(t) = y₁(t) + y₂(t), and substitute it into the differential equation:

t⁵(y₁(t) + y₂(t))' - t³(y₁(t) + y₂(t))'' + 6(y₁(t) + y₂(t)) = ?

Using the properties of derivatives, we can rewrite this as:

t⁵(y₁'(t) + y₂'(t)) - t³(y₁''(t) + y₂''(t)) + 6(y₁(t) + y₂(t)) = ?

Distributing the terms, we get:

t⁵y₁'(t) + t⁵y₂'(t) - t³y₁''(t) - t³y₂''(t) + 6y₁(t) + 6y₂(t) = ?

Rearranging the terms, we obtain:

(t⁵y₁'(t) - t³y₁''(t) + 6y₁(t)) + (t⁵y₂'(t) - t³y₂''(t) + 6y₂(t)) = ?

Since y₁(t) and y₂(t) are solutions, we know that each term in parentheses equals zero:

0 + 0 = 0

Thus, the sum y₁(t) + y₂(t) also satisfies the differential equation, confirming the additivity principle.

Homogeneity

Now, let's examine the homogeneity principle. Suppose y(t) is a solution to the differential equation:

t⁵y'(t) - t³y''(t) + 6y(t) = 0

Let c be any constant, and consider the function cy(t). Substituting cy(t) into the differential equation, we get:

t⁵(cy(t))' - t³(cy(t))'' + 6(cy(t)) = ?

Using the properties of derivatives, we can rewrite this as:

t⁵(cy'(t)) - t³(cy''(t)) + 6cy(t) = ?

Factoring out the constant c, we obtain:

c(t⁵y'(t) - t³y''(t) + 6y(t)) = ?

Since y(t) is a solution, the term in parentheses equals zero:

c(0) = 0

Therefore, cy(t) also satisfies the differential equation, confirming the homogeneity principle.

Conclusion on Linearity

Since the given differential equation satisfies both the additivity and homogeneity principles, we can definitively conclude that it is a linear differential equation.

Determining Order

To determine the order of the differential equation, we simply need to identify the highest order derivative present in the equation. In the equation t⁵y' - t³y'' + 6y = 0, we observe the following derivatives:

  • y' represents the first derivative of y(t).
  • y'' represents the second derivative of y(t).

The highest order derivative present is y'', which is the second derivative. Therefore, the order of the differential equation is 2.

Final Answer: Linearity and Order of the Differential Equation

Based on our comprehensive analysis, we can conclude that the differential equation t⁵y' - t³y'' + 6y = 0 is:

  • Linear
  • Of second order

Therefore, the correct answer is (D) Linear de quarta ordem.

This detailed analysis demonstrates the process of determining the linearity and order of a differential equation, providing a framework for analyzing a wide range of differential equations encountered in various scientific and engineering disciplines. Understanding these fundamental properties is crucial for selecting appropriate solution techniques and interpreting the behavior of dynamic systems modeled by differential equations.