Solving Dy/dx = 3x² / (1 + Y²) Differential Equation With Separation Of Variables

by BRAINLY IN FTUNILA 82 views
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Differential equations are fundamental tools in various fields of science and engineering, modeling phenomena that change over time or space. One common type of differential equation is the separable equation, where the variables can be isolated on opposite sides of the equation. This article delves into the method of separation of variables to solve the differential equation dy/dx = 3x² / (1 + y²). We'll break down the steps, provide a detailed explanation, and highlight the underlying concepts to ensure a comprehensive understanding.

Understanding Separable Differential Equations

Before diving into the solution, it's crucial to grasp the concept of separable differential equations. A differential equation is considered separable if it can be written in the form f(y) dy = g(x) dx, where f(y) is a function of y only and g(x) is a function of x only. This separation allows us to integrate both sides independently, leading to a solution that expresses the relationship between x and y.

In our case, the given equation dy/dx = 3x² / (1 + y²) fits this criterion. We can manipulate it algebraically to isolate the y terms on one side and the x terms on the other. This is the essence of the separation of variables technique, allowing us to transform a complex differential equation into two simpler integrals.

Step-by-Step Solution

Let's walk through the solution step-by-step:

1. Separate the Variables

The first step in solving a separable differential equation involves rearranging the terms to isolate the variables. To separate the variables in the equation dy/dx = 3x² / (1 + y²), we multiply both sides by (1 + y²) and by dx. This results in the following equation:

(1 + y²) dy = 3x² dx

Now, all the terms involving y are on the left side, and all the terms involving x are on the right side. This separation is the foundation for the integration step that follows. This strategic algebraic manipulation transforms the differential equation into a form where standard integration techniques can be applied, making the problem significantly more manageable. Separating variables is a crucial first step in solving differential equations of this type, as it sets the stage for finding the general solution.

2. Integrate Both Sides

With the variables separated, the next step is to integrate both sides of the equation. Integrating (1 + y²) dy with respect to y and 3x² dx with respect to x, we get:

∫ (1 + y²) dy = ∫ 3x² dx

Evaluating these integrals yields:

y + (y³/3) = x³ + C

where C is the constant of integration. This constant arises because the derivative of a constant is zero, meaning there are infinitely many possible constant terms that could be present in the original function. The constant of integration, C, represents the family of solutions to the differential equation. Each value of C corresponds to a unique solution curve. Determining the value of C often requires additional information, such as an initial condition, which specifies the value of y at a particular value of x. The integration step is pivotal as it transforms the differential equation into an algebraic equation, albeit one that includes an arbitrary constant. This algebraic equation represents the general solution to the differential equation.

3. General Solution

The equation y + (y³/3) = x³ + C represents the general solution to the given differential equation. This solution implicitly defines the relationship between y and x. It's called the general solution because it includes the arbitrary constant C, which can take on any value. Each specific value of C corresponds to a particular solution curve. The general solution encapsulates all possible solutions to the differential equation. To find a specific solution, we need additional information, such as an initial condition. An initial condition is a point (x₀, y₀) that lies on the solution curve. Substituting these values into the general solution allows us to solve for C and obtain a unique solution. The general solution provides a comprehensive representation of the behavior of the system described by the differential equation, while specific solutions provide precise predictions for particular scenarios.

4. Implicit vs. Explicit Solutions

The solution y + (y³/3) = x³ + C is an implicit solution. This means that y is not explicitly expressed as a function of x. To obtain an explicit solution, we would need to solve the equation for y. However, in this case, solving for y is a cubic equation, which can be complex. In many situations, an implicit solution is perfectly acceptable and provides valuable information about the relationship between the variables. An explicit solution, if obtainable, offers a more direct way to compute y for a given x, but it may not always be feasible or necessary. Understanding the distinction between implicit and explicit solutions is crucial in the context of differential equations, as it guides the approach to solving and interpreting the results. In some cases, numerical methods may be employed to approximate solutions when an explicit solution is difficult or impossible to find. The choice between seeking an implicit or explicit solution depends on the specific problem and the desired level of detail and computational ease.

Importance of Initial Conditions

The general solution y + (y³/3) = x³ + C represents a family of solutions, each corresponding to a different value of C. To find a unique solution, we need an initial condition, which is a point (x₀, y₀) that the solution curve must pass through. Substituting x₀ and y₀ into the general solution allows us to solve for C and obtain a particular solution.

For instance, suppose we have the initial condition y(0) = 1. This means that when x = 0, y = 1. Substituting these values into the general solution, we get:

1 + (1³/3) = 0³ + C

4/3 = C

Therefore, the particular solution that satisfies the initial condition is:

y + (y³/3) = x³ + 4/3

Initial conditions play a vital role in determining the specific solution that accurately models a given physical system or phenomenon. They provide the necessary constraint to select one solution from the infinite family of solutions represented by the general solution. Without initial conditions, we can only describe the general behavior of the system, but we cannot pinpoint its exact state at a particular time or location. The concept of initial conditions is fundamental in the application of differential equations to real-world problems, as it allows us to make concrete predictions and understand the specific evolution of a system under given circumstances. The integration constant, C, is the gateway through which initial conditions influence the particular solution, tailoring the general solution to fit the specific context of the problem.

Applications and Significance

The differential equation dy/dx = 3x² / (1 + y²), while seemingly abstract, can model various real-world phenomena. Differential equations, in general, are used extensively in physics, engineering, biology, and economics to describe rates of change and dynamic systems.

For example, this equation could represent the rate of change of a quantity y with respect to another quantity x, where the rate of change is influenced by both x and y themselves. The specific interpretation would depend on the context of the problem. Understanding how to solve separable differential equations is a crucial skill for anyone working in these fields.

In physics, differential equations are used to model motion, heat transfer, and wave propagation. In engineering, they are essential for designing circuits, analyzing structural stability, and controlling systems. In biology, they can model population growth, the spread of diseases, and chemical reactions. In economics, they are used to analyze market trends, model economic growth, and optimize investment strategies. The ability to formulate and solve differential equations is a cornerstone of quantitative analysis in these disciplines. The technique of separation of variables, as demonstrated in this article, is a powerful tool for tackling a wide range of differential equations that arise in these diverse fields. By mastering this technique, students and professionals can gain a deeper understanding of the world around them and develop the ability to make informed predictions and decisions.

Conclusion

In this article, we have demonstrated how to solve the differential equation dy/dx = 3x² / (1 + y²) using the method of separation of variables. This technique involves isolating the variables on opposite sides of the equation, integrating both sides, and obtaining a general solution. We also highlighted the importance of initial conditions in determining a particular solution. Separable differential equations are a fundamental concept in calculus and have wide-ranging applications in various fields. By mastering this technique, you can tackle a wide range of problems involving rates of change and dynamic systems. The ability to solve differential equations is a valuable asset for anyone pursuing studies or a career in science, engineering, or other quantitative disciplines. The concepts and techniques presented here provide a solid foundation for further exploration of more advanced topics in differential equations and their applications. The journey of understanding differential equations is a continuous process of learning and applying these powerful tools to unravel the complexities of the world around us. The power to model and predict the behavior of dynamic systems is a skill that opens doors to countless opportunities in research, innovation, and problem-solving across a multitude of domains.