Problem 18 - Advanced Engineering Mathematics

Question

For \(y^{\prime}+(\cot x) y=\sec ^{2} x \csc x\) an integrating factor is \(e^{\int \cot x d x}=e^{\ln |\sin x|}=\sin x\) so that \(\frac{d}{d x}[(\sin x) y]=\sec ^{2} x\) and \(y=\sec x+c \csc x\) for \(0< x<\pi / 2\).

Step-by-Step Solution

Answer

The solution to the differential equation is \(y = \sec x + C \csc x\) for \(0 < x < \pi/2\).

Step 1: Identify the Given Differential Equation

The given differential equation is first-order linear: \( y' + ( an x) y = \sec^2 x \csc x \). It is of the form \( y' + P(x)y = Q(x) \) where \( P(x) = \tan x \) and \( Q(x) = \sec^2 x \csc x \).

Step 2: Determine the Integrating Factor

The integrating factor \( \mu(x) \) is defined as \( e^{\int P(x) \, dx} \). Here, \( P(x) = \cot x \), so \( \mu(x) = e^{\int \cot x \, dx} = e^{\ln |\sin x|} = \sin x \).

Step 3: Multiply the Differential Equation by the Integrating Factor

To make the left side of the equation exact, multiply the entire differential equation by \( \sin x \), resulting in: \( \sin x \cdot y' + \sin x \cdot (\cot x) y = \sin x \cdot \sec^2 x \csc x \). This simplifies to \( \frac{d}{dx}[(\sin x)y] = \sec^2 x \).

Step 4: Integrate Both Sides

Integrate both sides with respect to \( x \): \( \int \frac{d}{dx}[(\sin x)y] \, dx = \int \sec^2 x \, dx \). The left side simplifies to \( (\sin x)y \) and the right side integrates to \( \tan x + C \).

Step 5: Solve for \( y \)

Re-arrange the equation \( (\sin x)y = \tan x + C \) to solve for \( y \). Divide both sides by \( \sin x \): \( y = \frac{\tan x + C}{\sin x} \). Recognizing trigonometric identities, \( y = \sec x + C \csc x \).

Step 6: Apply Constraints

Apply the given constraints \( 0 < x < \pi/2 \) to ensure the solution is valid, implying \( \sin x > 0 \), thus the solution can be expressed as \( y = \sec x + C \csc x \).

Key Concepts

Integrating Factor

The concept of an integrating factor is essential when dealing with first-order linear differential equations. It is a function we use to transform a non-exact differential equation into an exact one, which is easier to solve. To find the integrating factor, we need to first identify the differential equation in the standard form of \( y' + P(x)y = Q(x) \). After identifying \( P(x) \), the integrating factor is given by \( \mu(x) = e^{\int P(x) \, dx} \). Let's break this down further:

• Integrate \( P(x) \) over \( x \), which in our exercise is \( \cot x \). The integral of \( \cot x \) is \( \ln |\sin x| \).
• Substitute this back into the exponential, giving us \( e^{\ln |\sin x|} = \sin x \).

Now, you have the integrating factor \( \sin x \), which is a tool to make the equation into an easily solvable exact form.

Trigonometric Integration

Trigonometric integration refers to integrating functions that involve trigonometric identities or expressions. In this exercise, once we have our differential equation multiplied by the integrating factor, the next step involves integrating: - The left-hand side becomes \( \frac{d}{dx}[(\sin x)y] \). - The right-hand side involves integrating \( \sec^2 x \).
For trigonometric functions like \( \sec^2 x \), it is useful to remember the standard derivatives and integrals:

• For instance, the integral of \( \sec^2 x \) is a basic trigonometric integral recognized as \( \tan x \).

Thus, by performing this integral, we end up with \( \tan x + C \) for the right side, simplifying the process of solving for \( y \). Trigonometric integration usually involves knowing key integrals and being able to apply them with algebraic manipulation.

Exact Equation

An exact equation is one where the total derivative of a function equals a given expression, meaning it can be directly integrated. For first-order linear differential equations, making the left-hand side an exact derivative is crucial for integration. In our scenario, once multiplied by the integrating factor, we obtain an exact equation:

• The differential form \( \sin x \cdot y' + \sin x \cdot (\cot x) y = \sec^2 x \cdot \csc x \) becomes \( \frac{d}{dx}[(\sin x)y] = \sec^2 x \).

The process ensures that the function \((\sin x)y\) behaves exactly in the derivative sense, thereby allowing for straightforward integration with respect to \( x \). Thus, the equation is a bridge to move directly towards the solution by solving for the original function inside the derivative.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that hold true for all angles. These identities are critical in simplifying and solving trigonometric equations encountered in differential forms. For example, during the process of solving for \( y \), we see:

• The result of \( y = \frac{\tan x + C}{\sin x} \) can be rewritten using known identities.
  
• The expression transforms to \( y = \sec x + C \csc x \), using identities like \( \tan x = \frac{\sin x}{\cos x} \) and \( \sec x = \frac{1}{\cos x} \).
  

Identities like \( \sec^2 x = 1 + \tan^2 x \) provide shortcuts in integration and expression simplification. Recognizing and applying these identities is vital for obtaining solutions efficiently, especially in trigonometric integration and manipulation.