Problem 11 - Advanced Engineering Mathematics

Question

In Problems 1-22, solve the given differential equation by separation of variables. $$ \csc y d x+\sec ^{2} x d y=0 $$

Step-by-Step Solution

Answer

The general solution is \( \frac{x}{2} + \frac{1}{4} \sin(2x) + \cos y = C \).

Step 1: Rewrite the Equation

Start by rewriting the given differential equation to separate the variables. The original equation is \( \csc y \; dx + \sec^2 x \; dy = 0 \). We want to rearrange it so that all terms involving \( x \) are on one side and all terms involving \( y \) are on the other side. Subtract \( \sec^2 x \; dy \) from both sides to get: \( \csc y \; dx = -\sec^2 x \; dy \).

Step 2: Separate the Variables

Divide both sides by \( \csc y \cdot \sec^2 x \) to separate the variables. This gives us: \( \frac{dx}{\sec^2 x} + \frac{dy}{\csc y} = 0 \). Simplify this to \( \cos^2 x \, dx + \sin y \, dy = 0 \). Now, each term involves only one variable.

Step 3: Integrate Both Sides

Integrate each term separately. For the \( x \) term: \( \int \cos^2 x \, dx \) and for the \( y \) term: \( \int \sin y \, dy \). Recall that \( \int \cos^2 x \, dx = \frac{1}{2}x + \frac{1}{4}\sin(2x) + C_x\) using the double angle identity, and \( \int \sin y \, dy = -\cos y + C_y \). Simplifying, you get: \( \frac{1}{2}x + \frac{1}{4}\sin(2x) + C_x + \cos y - C_y = 0 \).

Step 4: Solve for the General Solution

Combine the constants \( C_x \) and \( -C_y \) into a single constant \( C \), as \( C_x - C_y = C \). Thus, the equation becomes: \( \frac{x}{2} + \frac{1}{4} \sin(2x) + \cos y = C \). Solve for \( y \) if necessary depending on additional conditions or initial values.

Key Concepts

Separation of Variables

Separation of Variables is a method used to solve differential equations. This approach is particularly useful when you can rearrange an equation such that each variable and its differential are isolated on opposite sides of the equation. In this problem, the original equation \[ \csc y \ dx + \sec^2 x \ dy = 0 \]must be rearranged. Here, separation begins by moving all the terms involving \( y \) to one side and all terms involving \( x \) to the other. This results in:\[ \csc y \ dx = -\sec^2 x \ dy \]To completely separate the variables, divide each side by the product of their counterparts, yielding:\[ \frac{dx}{\sec^2 x} + \frac{dy}{\csc y} = 0 \]This transforms the equation into a form where direct integration is possible. Understanding this method enables you to solve the differential equation by integrating each side independently.

Integration

Integration is a fundamental concept often required once the variables in a differential equation are separated. In this equation, \[ \cos^2 x \, dx + \sin y \, dy = 0 \]we need to integrate each term.

• Integrating \( \cos^2 x \): We use the trigonometric identity \( \cos^2 x = \frac{1 + \cos(2x)}{2} \), which simplifies the integral to: \[ \int \cos^2 x \, dx = \frac{1}{2}x + \frac{1}{4}\sin(2x) + C_x \]


• Integrating \( \sin y \): This simple integral results in: \[ \int \sin y \, dy = -\cos y + C_y \]

By integrating both sides, we find an expression with a general constant that can later be adjusted based on initial conditions or boundary conditions.

Trigonometric Functions

During the process of separating variables and integrating, the use of trigonometric functions becomes crucial. Here are some essential trigonometric insights:

• \( \cos^2 x \) is often expressed using identities like \( \cos^2 x = \frac{1 + \cos(2x)}{2} \), due to which it simplifies integration tasks.


• Understanding identities helps to transform complex trigonometric functions into simpler, either by converting them into polynomials or other recognizable forms.


• \( \int \cos^2 x \, dx \) and \( \int \sin y \, dy \) highlight the necessity to recognize known integrals and apply rules seamlessly.


• Such functions are not only vital in calculations but also serve as tools for identifying and solving integration bounds.

By mastering the use of trigonometric functions within integrals and equations, solving complex equations becomes manageable and straightforward.