Solving Cos(x)cos(y)dy/dx + (sin(x) - Cos(x))sin(y) = 0 A Step-by-Step Guide

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In this article, we will delve into the process of finding the general solution of the given differential equation: cos(x)cos(y)dy/dx + (sin(x) - cos(x))sin(y) = 0. This equation falls under the category of first-order differential equations, and we will employ techniques such as separation of variables to arrive at the solution. Understanding how to solve such equations is crucial in various fields of science and engineering, as they often model real-world phenomena. Differential equations are mathematical equations that relate a function with its derivatives. In various disciplines like physics, engineering, economics, and biology, these equations play a vital role in modeling dynamic systems and processes. Solving a differential equation means finding the function that satisfies the equation. There are several methods for solving differential equations, including analytical methods, numerical methods, and qualitative methods. The choice of method depends on the type and complexity of the equation, as well as the desired accuracy and efficiency of the solution.

Our objective is to determine the general solution of the following differential equation:

cos(x)cos(y)dy/dx + (sin(x) - cos(x))sin(y) = 0

This equation represents a relationship between the functions y(x) and its derivative dy/dx, along with trigonometric functions of x and y. To solve this, we will manipulate the equation to separate the variables and then integrate both sides.

The method of separation of variables is a powerful technique for solving first-order differential equations, especially those that can be written in a form where the terms involving the dependent variable (y) and its derivative can be isolated on one side of the equation, while the terms involving the independent variable (x) are on the other side. This method simplifies the equation, allowing us to integrate each side separately and then combine the results to find the general solution. In this case, we will rearrange the given equation to group y terms with dy and x terms with dx, making the equation amenable to integration. This technique is fundamental in solving various types of differential equations encountered in physics, engineering, and other scientific disciplines. The separation of variables is a method to solve differential equations by isolating the variables on different sides of the equation. This method is applicable when the differential equation 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. By separating the variables, we can integrate both sides of the equation independently, which often leads to a simpler solution process. This method is particularly useful for first-order ordinary differential equations and is widely used in various fields, including physics, engineering, and mathematics, to model and solve problems involving rates of change.

Step-by-Step Solution

  1. Rearrange the Equation: Begin by moving the term (sin(x) - cos(x))sin(y) to the right side of the equation: cos(x)cos(y)dy/dx = -(sin(x) - cos(x))sin(y)

  2. Separate the Variables: Divide both sides by cos(x)sin(y) to separate y terms on the left and x terms on the right: (cos(y)dy) / sin(y) = -(sin(x) - cos(x))dx / cos(x)

  3. Simplify the Equation: Rewrite the equation to make it easier to integrate: (cos(y) / sin(y))dy = -(sin(x) / cos(x) - 1)dx cot(y)dy = -(tan(x) - 1)dx

  4. Integrate Both Sides: Integrate both sides of the equation with respect to their respective variables: ∫cot(y)dy = ∫(1 - tan(x))dx

    The integral of cot(y) is ln|sin(y)|, and the integral of (1 - tan(x)) is x + ln|cos(x)|. Therefore, ln|sin(y)| = x + ln|cos(x)| + C where C is the constant of integration. The integral of cot(y) can be found using the substitution method. Let u = sin(y), then du = cos(y) dy. Thus, the integral becomes ∫(1/u) du, which is ln|u| + C, or ln|sin(y)| + C. Similarly, the integral of tan(x) can be found by rewriting it as sin(x)/cos(x) and using the substitution method. Let v = cos(x), then dv = -sin(x) dx. The integral becomes -∫(1/v) dv, which is -ln|v| + C, or -ln|cos(x)| + C. Combining these integrals allows us to proceed with solving the differential equation. Understanding the techniques for integrating trigonometric functions is crucial for solving many differential equations that arise in physics and engineering. By mastering these methods, we can effectively analyze and solve a wide range of problems involving rates of change and dynamic systems. The constant of integration, C, is added because there are infinitely many solutions that differ by a constant. This constant is determined by the initial conditions of the problem, if given. In the absence of initial conditions, we leave the solution in terms of C, representing the general solution to the differential equation.

  5. Solve for y: Exponentiate both sides to eliminate the natural logarithm: |sin(y)| = e^(x + ln|cos(x)| + C) |sin(y)| = e^x * e^(ln|cos(x)|) * e^C |sin(y)| = e^x * |cos(x)| * e^C

    Let K = ±e^C, where K is another constant. We can rewrite the equation as: sin(y) = K * e^x * cos(x)

    Finally, solve for y: y = arcsin(K * e^x * cos(x)) The general solution to the given differential equation is y(x) = arcsin(Keˣcos(x)), where K is an arbitrary constant.

To ensure the correctness of our solution, we can verify it by substituting it back into the original differential equation. This involves differentiating the solution y(x) with respect to x and then plugging both y(x) and dy/dx into the original equation. If the equation holds true, then our solution is correct. This step is crucial in confirming that we have indeed found the general solution and that there are no errors in our calculations or manipulations. The process of verification not only validates the solution but also enhances our understanding of the differential equation and the techniques used to solve it. By carefully checking each step, we can build confidence in our result and ensure that it accurately represents the behavior of the system or phenomenon being modeled. In addition, verification helps in identifying any potential issues or limitations of the solution, such as specific intervals or conditions under which the solution is valid. This comprehensive approach to solving and verifying differential equations is essential for both theoretical and practical applications.

  1. Differentiate the Solution: Differentiate y = arcsin(K * e^x * cos(x)) with respect to x: To find dy/dx, we use the chain rule. Recall that the derivative of arcsin(u) is 1 / √(1 - u²) and apply the product rule to differentiate K * e^x * cos(x). Let u = K * e^x * cos(x), so

    du/dx = K(e^xcos(x) - e^xsin(x))

    Then, dy/dx = (1 / √(1 - (K * e^x * cos(x))²)) * K(e^xcos(x) - e^xsin(x))

  2. Substitute into the Original Equation: Plug y and dy/dx into the original equation: cos(x)cos(arcsin(K * e^x * cos(x))) * [(1 / √(1 - (K * e^x * cos(x))²)) * K(e^xcos(x) - e^xsin(x))] + (sin(x) - cos(x))sin(arcsin(K * e^x * cos(x))) = 0

    This simplifies to: cos(x) * √(1 - (K * e^x * cos(x))²) * [(1 / √(1 - (K * e^x * cos(x))²)) * K(e^xcos(x) - e^xsin(x))] + (sin(x) - cos(x)) * (K * e^x * cos(x)) = 0

    cos(x) * K * e^x * (cos(x) - sin(x)) + (sin(x) - cos(x)) * K * e^x * cos(x) = 0

    K * e^x * cos(x) * (cos(x) - sin(x)) - K * e^x * cos(x) * (cos(x) - sin(x)) = 0

    0 = 0

    Since the equation holds true, our solution is verified.

We have successfully found the general solution of the differential equation cos(x)cos(y)dy/dx + (sin(x) - cos(x))sin(y) = 0 using the method of separation of variables. The general solution is given by:

y(x) = arcsin(Keˣcos(x)), where K is an arbitrary constant.

This solution satisfies the original differential equation, as verified by substituting it back into the equation. The separation of variables technique is a powerful tool for solving many first-order differential equations, and this example demonstrates its application in detail. Understanding and mastering these techniques is crucial for solving more complex problems in various fields of science and engineering. In summary, solving differential equations is a fundamental skill in mathematics and its applications. The method of separation of variables provides an effective way to solve first-order differential equations that can be written in a separable form. By understanding the steps involved and practicing with different examples, one can develop a strong foundation in this area. The ability to solve differential equations allows us to model and analyze dynamic systems, making predictions and gaining insights into the behavior of the systems over time.