Problem 17 - Advanced Engineering Mathematics

Question

\text { For } y^{\prime}+(\tan x) y=\sec x \text { an integrating factor is } e^{\int \tan x d x}=\sec x \text { so that } \frac{d}{d x}[(\sec x) y]=\sec ^{2} x \text { and } \(y=\sin x+c \cos x\) for \(-\pi / 2< x<\pi / 2\).

Step-by-Step Solution

Answer

The solution is \( y = \sin(x) + c \cos(x) \) for \( -\pi/2 < x < \pi/2 \).

Step 1: Understanding the Differential Equation

We start with the first order linear differential equation: \( y' + \tan(x) y = \sec(x) \). Our task is to solve this equation using an integrating factor.

Step 2: Finding the Integrating Factor

The integrating factor for the differential equation \( y' + P(x) y = Q(x) \) is given by \( e^{\int P(x) \, dx} \). In this equation, \( P(x) = \tan(x) \), so the integrating factor becomes \( e^{\int \tan(x) \, dx} = \sec(x) \).

Step 3: Rewriting the Equation with the Integrating Factor

Multiplying every term in the equation by the integrating factor \( \sec(x) \), we get: \( \sec(x) y' + \sec(x) \tan(x) y = 1 \). This simplifies to: \( \frac{d}{dx}[\sec(x) y] = \sec^2(x) \).

Step 4: Integrating Both Sides

To solve for \( y \), integrate both sides with respect to \( x \). The left-hand side becomes \( \sec(x) y \), and the integral of \( \sec^2(x) \) is \( \tan(x) \). So, we have: \( \sec(x) y = \tan(x) + C \), where \( C \) is the constant of integration.

Step 5: Solving for \( y \)

To solve for \( y \), divide both sides by \( \sec(x) \), yielding: \( y = \sin(x) + C \cos(x) \). Here, \( \sec(x) = 1/\cos(x) \), and \( \tan(x) = \sin(x)/\cos(x) \), hence, \( \tan(x)+C = \sin(x) + C\cos(x) \).

Key Concepts

Integrating Factor

When dealing with first order linear differential equations, the integrating factor is a clever mathematical tool used to make the equations easier to solve. To apply this method, you first need to rewrite the differential equation in standard form, i.e., \( y' + P(x)y = Q(x) \). The function \( P(x) \) is then used to determine the integrating factor.

The integrating factor is calculated as \( e^{\int P(x) \, dx} \). In the example given, for the equation \( y' + \tan(x)y = \sec(x) \), \( P(x) = \tan(x) \). Therefore, the integrating factor becomes \( e^{\int \tan(x) \, dx} = \sec(x) \).

Once you find the integrating factor, you multiply every term in the differential equation by this factor. This simplifies the equation and allows you to express it as the derivative of a product, making it easier to solve.

First Order Linear Differential Equation

A first order linear differential equation is characterized by an equation of the form \( y' + P(x)y = Q(x) \), where \( y' \) is the derivative of \( y \) with respect to \( x \), and \( P(x) \) and \( Q(x) \) are functions of \( x \). These equations are called "linear" because the unknown function \( y \) and its derivative appear in linear form, i.e., to the first power.

Solving a first order linear differential equation often involves finding an integrating factor. Once you multiply through by the integrating factor, the left-hand side of the equation becomes the derivative of a product of two functions. This reduction simplifies the solving process initially set up to find \( y(x) \). Such equations commonly appear in modeling processes where some rate of change depends linearly on one particular variable and its rate of change.

These equations are integral to understanding many real-world phenomena, from physical systems to population dynamics.

Solution Techniques for Differential Equations

When solving differential equations, several techniques may be employed depending on the type of equation. For first order linear differential equations, one common approach is the method of integrating factors.

After identifying and applying the integrating factor, the differential equation can be rewritten as a derivative of a product, which simplifies the problem. This step allows for the use of basic calculus methods such as integration to find the solution.

The process follows these general steps:

• Identify \( P(x) \) in the equation \( y' + P(x)y = Q(x) \).
• Calculate the integrating factor \( e^{\int P(x) \, dx} \).
• Multiply the entire differential equation by this integrating factor.
• Express the equation in terms of a derivative of a product and integrate both sides.
• Solve for the function \( y(x) \).

Using these steps, the problem becomes more manageable and solvable with basic integration, highlighting the power and utility of integrating factors when dealing with such equations.