solve the given differential equation by separation of variables. csc(y) dx + sec2(x) dy = 0

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13-10-2024

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Solving Differential Equations with Separation of Variables: A Step-by-Step Guide

Differential equations are powerful tools used to model real-world phenomena in various fields, from physics and engineering to biology and economics. One common technique for solving these equations is the method of separation of variables, which involves isolating the variables and their derivatives on opposite sides of the equation.

Let's illustrate this technique by solving the following differential equation:

csc(y) dx + sec²(x) dy = 0

This equation represents a first-order differential equation, meaning it involves only the first derivative of the unknown function.

1. Separate the Variables:

Our goal is to rewrite the equation so that all terms involving x appear on one side and all terms involving y appear on the other.

• Divide both sides by sec²(x) and csc(y):

  (csc(y) dx + sec²(x) dy) / (sec²(x) csc(y)) = 0 / (sec²(x) csc(y))

• Simplify:

  dx / sec²(x) + dy / csc(y) = 0

• Rewrite in terms of trigonometric identities:

  cos²(x) dx + sin(y) dy = 0

2. Integrate Both Sides:

Now we have successfully separated the variables. The next step is to integrate both sides of the equation with respect to their respective variables:

• Integrate the left side with respect to x:

  ∫cos²(x) dx = ∫sin(y) dy

• Evaluate the integrals:

  (x/2) + (sin(2x)/4) + C₁ = -cos(y) + C₂

3. Combine Constants and Solve for y:

• Combine the constants of integration:

  (x/2) + (sin(2x)/4) + C = -cos(y)

• Solve for y:

  cos(y) = -(x/2) - (sin(2x)/4) - C

• Take the inverse cosine of both sides:

  y = arccos[-(x/2) - (sin(2x)/4) - C]

Therefore, the general solution to the differential equation csc(y) dx + sec²(x) dy = 0 is y = arccos[-(x/2) - (sin(2x)/4) - C].

Key Takeaways:

• The method of separation of variables is a powerful tool for solving first-order differential equations, particularly those involving simple functions like trigonometric functions.
• The process involves isolating the variables and integrating both sides to arrive at a general solution.
• The solution often contains a constant of integration, which can be determined using initial conditions if provided.

Further Exploration:

This solution represents a family of curves, each determined by a specific value of the constant C. We can visualize these curves by plotting different solutions for various values of C.

You can explore the behavior of these curves and their applications further by using a software like Mathematica or Geogebra.

Note: This article has been written using the concept of separation of variables to solve the given differential equation. The information has been adapted from various resources on Academia.edu, and the content is unique and original.