91Ó°ÊÓ

A first-order linear differential equation involves derivatives of the first order and can be written in the form: \ \ \[ \frac{dy}{dx} + P(x)y = Q(x) \]. \ \ This structure means that the equation includes a term with the derivative of the dependent variable (\( \frac{dy}{dx} \)) plus a term that is a function of both the dependent variable \(y\) and the independent variable \(x\), which is usually represented by \(P(x)y\). \ \ The given problem fits this form as: \ \ \[ \sin x \frac{dy}{dx} + 2 y \cos x = 1 \]. \ \ By dividing by \( \sin x \), we can isolate \( \frac{dy}{dx} \): \ \ \[ \frac{dy}{dx} + 2y \cot x = \csc x \]. \ \ Here, \( P(x) = 2 \cot x \) and \( Q(x) = \csc x \).

One efficient method for solving first-order linear differential equations is by using an integrating factor. The integrating factor helps convert the differential equation into a simpler form, often a differential of a product. \ \ To build the integrating factor \( \mu(x) \), follow these key steps: \ \ 1. Identify \( P(x) \) from the rearranged equation. \ \ 2. Compute the integral of \( P(x) \). \ \ 3. Calculate the exponential: \[ \mu(x) = e^{\int P(x) \, dx} \]. \ \ For the given equation, \( P(x) = 2 \cot x \). Integrating it: \ \ \[ \mu(x) = e^{\int 2 \cot x \, dx} = e^{2 \ln|\sin x|} = (\sin x)^2 \].

Boundary conditions are constraints provided in a differential equation to determine a unique solution. They specify the value of the dependent variable \( y \) at a particular point of the independent variable \( x \). \ \ For the given equation, we have the boundary condition: \ \ \[ y \left( \frac{\pi}{2} \right) = 1 \]. \ \ To integrate the converted differential equation, we use this condition to solve for the constant of integration \( C \), ensuring the solution is uniquely determined. After integrating both sides: \ \ \[ y(\sin x)^2 = -\cos x + C \]. \ \ By substituting the boundary condition into this equation: \ \ \[ 1 (\sin \frac{\pi}{2})^2 = -\cos \frac{\pi}{2} + C \]. \ \ Simplify: \( 1 = 0 + C \). Thus, \( C = 1 \).

Variable separation is a technique used to rearrange a differential equation so that each side contains only one variable. However, not all linear differential equations can be separated directly. \ \ In the given problem, variable separation was used indirectly. By dividing the original equation: \ \ \[ \sin x \frac{dy}{dx} + 2y \cos x = 1 \] \ \ by \( \sin x \), we separated variables initially: \ \ \[ \frac{dy}{dx} + 2y \cot x = \csc x \]. \ \ Next, the integrating factor was applied to convert the differential equation into a form that allowed for integration through the variable separation method: \ \ \[ \frac{d}{dx} [y (\sin x)^2] = \sin x \]. \ \ When you see equations formulated in this way, identify if variable separation can simplify the integration process.