Solving The Differential Equation 3f'(x) + 6/(2x-6)^3 = 0 A Step-by-Step Guide
In this article, we will delve into the process of solving the differential equation 3f'(x) + 6/(2x-6)^3 = 0. Differential equations play a crucial role in various fields of science and engineering, as they model systems that change over time or space. Finding solutions to these equations allows us to understand and predict the behavior of these systems. This particular equation is a first-order differential equation, and we will employ techniques of separation of variables and integration to arrive at the general solution. Understanding how to solve such equations is fundamental in many areas, from physics and chemistry to economics and computer science. This exploration will not only provide a step-by-step guide to solving this specific equation but also enhance the understanding of general methods applicable to a broader range of differential equations. We will explore the underlying principles and mathematical manipulations required to arrive at the solution, ensuring a thorough and clear understanding of the process.
Understanding Differential Equations
Before diving into the solution, let’s briefly discuss what a differential equation is and why they are important. A differential equation is an equation that relates a function with its derivatives. In simpler terms, it's an equation where the unknown is a function, and the equation includes the function's rate of change (derivatives). These equations are powerful tools for modeling phenomena involving change, such as population growth, radioactive decay, and the motion of objects. The order of a differential equation is determined by the highest derivative present in the equation. In our case, we are dealing with a first-order differential equation because it involves only the first derivative, f'(x). The solutions to a differential equation are not unique; they form a family of functions that satisfy the equation. This family is often represented by a general solution that includes an arbitrary constant, usually denoted by C. This constant accounts for the different possible initial conditions or boundary conditions that might apply to the system being modeled. Solving differential equations is a cornerstone of many scientific disciplines, enabling us to make predictions and design systems with desired behaviors. Therefore, a solid understanding of these equations and the techniques to solve them is invaluable for anyone pursuing studies or a career in science, technology, engineering, or mathematics (STEM) fields.
Rewriting the Differential Equation
The first step in solving our differential equation 3f'(x) + 6/(2x-6)^3 = 0 is to isolate the derivative term. This process involves algebraic manipulation to separate the terms involving f'(x) from the rest of the equation. By subtracting 6/(2x-6)^3 from both sides, we obtain: 3f'(x) = -6/(2x-6)^3. This step is crucial because it sets the stage for separating variables, a common technique used to solve first-order differential equations. Separation of variables involves rearranging the equation such that terms involving the dependent variable (f(x) in our case) are on one side and terms involving the independent variable (x in our case) are on the other side. Isolating the derivative also allows us to clearly see the relationship between the rate of change of the function and the expression involving x. This initial manipulation is not just a mechanical step; it’s a vital part of the problem-solving strategy, allowing us to apply integration techniques more effectively. By isolating f'(x), we're essentially preparing the equation for integration, which is the next step in finding the solution to the differential equation. This careful algebraic manipulation is a hallmark of solving differential equations and requires attention to detail to avoid errors that could lead to an incorrect solution.
Separating Variables
Now that we have isolated the derivative, the next crucial step is to separate the variables. This technique allows us to rearrange the equation so that each variable appears on only one side. Starting with 3f'(x) = -6/(2x-6)^3, we first divide both sides by 3 to simplify the equation, resulting in f'(x) = -2/(2x-6)^3. Recognizing that f'(x) is the derivative of f(x) with respect to x, we can rewrite it as df/dx. So, our equation becomes df/dx = -2/(2x-6)^3. The goal of separating variables is to get all terms involving f on one side and all terms involving x on the other. To achieve this, we multiply both sides of the equation by dx, which gives us df = -2/(2x-6)^3 dx. Now, all terms involving f are on the left side (df), and all terms involving x are on the right side (-2/(2x-6)^3 dx). This separation is a key step because it allows us to integrate both sides of the equation independently. The process of separating variables is fundamental in solving many types of differential equations, especially those that are first-order and separable. It transforms the original equation into a form that can be directly integrated, leading us closer to finding the general solution. Correct separation of variables is crucial; any error in this step will propagate through the rest of the solution process.
Integrating Both Sides
With the variables successfully separated, we can now integrate both sides of the equation. This step is the heart of solving the differential equation, as it allows us to find the original function f(x). We have the equation df = -2/(2x-6)^3 dx. Integrating both sides gives us ∫df = ∫-2/(2x-6)^3 dx. The left-hand side integral, ∫df, is straightforward and simply evaluates to f(x). The right-hand side integral, ∫-2/(2x-6)^3 dx, requires a bit more attention. To solve this integral, we can use a substitution method. Let u = 2x - 6, then du = 2 dx, and dx = du/2. Substituting these into the integral, we get ∫-2/u^3 (du/2), which simplifies to ∫-1/u^3 du. This integral can be rewritten as ∫-u^(-3) du. Now, we can apply the power rule for integration, which states that ∫x^n dx = (x^(n+1))/(n+1) + C, where C is the constant of integration. Applying this rule, we get -∫u^(-3) du = - (u^(-2)/(-2)) + C = 1/(2u^2) + C. Now, we substitute back u = 2x - 6, which gives us 1/(2(2x-6)^2) + C*. Therefore, the integral of the right-hand side is 1/(2(2x-6)^2) + C*. Combining the results from both sides, we have f(x) = 1/(2(2x-6)^2) + C. This integration step is crucial, and the correct application of integration techniques and the inclusion of the constant of integration are essential for obtaining the general solution of the differential equation. The constant of integration, C, represents the family of solutions to the differential equation, each differing by a constant vertical shift.
The General Solution
After performing the integration, we arrive at the general solution of the differential equation. From the previous step, we found that f(x) = 1/(2(2x-6)^2) + C. This equation represents the family of functions that satisfy the original differential equation 3f'(x) + 6/(2x-6)^3 = 0. The constant C is the constant of integration, and it is an arbitrary constant, meaning it can take any value. Each different value of C corresponds to a different solution curve. The presence of this constant is a key characteristic of the general solution to a differential equation. It reflects the fact that there are infinitely many functions whose derivatives satisfy the original equation. To find a specific, or particular, solution, we would need an initial condition, which is a value of f(x) at a specific x. The general solution provides the framework, and the initial condition allows us to pinpoint a single solution from this family. In the context of applications, the constant C often has a physical interpretation, representing an initial state or a background level of some quantity. For example, in a model of population growth, C might represent the initial population size. Understanding the general solution and the role of the constant of integration is essential for fully grasping the behavior of the system being modeled by the differential equation. The general solution provides a complete picture of all possible solutions, allowing us to analyze and predict the system's behavior under various conditions.
Verifying the Solution
To ensure the correctness of our solution, it is crucial to verify the solution by substituting it back into the original differential equation. Our proposed solution is f(x) = 1/(2(2x-6)^2) + C. The original differential equation is 3f'(x) + 6/(2x-6)^3 = 0. To verify, we first need to find the derivative f'(x). We can rewrite f(x) as f(x) = (1/2)(2x-6)^(-2) + C. Now, differentiating with respect to x, we use the chain rule: f'(x) = (1/2)(-2)(2x-6)^(-3)(2) = -2(2x-6)^(-3) = -2/(2x-6)^3. Now, we substitute f'(x) back into the original differential equation: 3(-2/(2x-6)^3) + 6/(2x-6)^3. This simplifies to -6/(2x-6)^3 + 6/(2x-6)^3, which equals 0. Since the equation holds true, our solution f(x) = 1/(2(2x-6)^2) + C is indeed a valid solution to the given differential equation. Verification is an essential step in solving differential equations and other mathematical problems. It provides a check against errors that might have occurred during the solution process. By substituting the solution back into the original equation, we can confirm that it satisfies the equation and that our calculations are correct. This step increases our confidence in the accuracy of the result and ensures that we have found the correct solution to the problem. In practical applications, verifying the solution is particularly important, as it ensures that the model we are using accurately represents the system being studied.
Conclusion
In summary, we have successfully solved the differential equation 3f'(x) + 6/(2x-6)^3 = 0. We started by isolating the derivative term, then separated the variables, integrated both sides, and arrived at the general solution f(x) = 1/(2(2x-6)^2) + C. Finally, we verified our solution by substituting it back into the original equation, confirming its validity. This process demonstrates a standard method for solving first-order separable differential equations. The key steps include algebraic manipulation to isolate the derivative, separating variables to get each variable on its own side, integrating both sides to find the function, and including the constant of integration to represent the family of solutions. Verification is a critical step to ensure the accuracy of the solution. Understanding these techniques is essential for anyone working with mathematical models in science, engineering, or other fields. Differential equations are fundamental tools for describing and predicting the behavior of dynamic systems, and the ability to solve them is a valuable skill. The specific equation we solved here is a relatively simple example, but the principles and methods we used are applicable to a wide range of more complex differential equations. Mastering these techniques lays the foundation for further study in differential equations and their applications. The solution we found, f(x) = 1/(2(2x-6)^2) + C, represents a family of functions, each differing by a constant, that satisfy the given differential equation. This constant, C, allows for different initial conditions or boundary conditions to be satisfied, making the general solution a versatile tool for modeling various situations.
Therefore, the correct answer is:
c
