91Ó°ÊÓ

The method of separation of variables is a popular technique for solving differential equations, particularly when variables can be relocated to different sides of the equation independently. By isolating each variable on opposite sides, integration can proceed smoothly. In the given exercise, the differential equation \( y^{\prime} \tan x = a + y \) was simplified by expressing \( y^{\prime} \) as \( \frac{a + y}{\tan x} \). Recognizing that \( \frac{1}{\tan x} \) is equivalent to \( \cot x \) allowed us to rewrite the equation as \( y^{\prime} = (a + y) \cot x \).
By further separating the variables, we rewrote the equation to \( \frac{dy}{a + y} = \cot x \, dx \). This transformation enabled us to perform integration on both sides independently, a necessary step for solving the equation using this technique. When using separation of variables, always verify that rearranged terms allow for direct integration.

Initial conditions are vital in determining a specific solution to a differential equation. They are values given for the dependent variable(s) at specific points, guiding the solution towards a unique path.
In our example, the initial condition is given as \( y(\pi/3) = a \). Having this information means that once we have integrated the equation and solved it generally for \( y \), we can find any constants by substituting these initial conditions back into the general solution. This step ensures a solution that uniquely passes through the known point. By applying \( x = \pi/3 \) and \( y = a \) to the expression \( a+y = C_1 \sin x \), and solving for \( C_1 \), we construct a particular solution: \( y = \frac{4a}{\sqrt{3}} \sin x - a \). This solution precisely fits the given conditions, illustrating the critical role of initial conditions in solving differential equations.

Integrating functions is a core step in solving differential equations once variables have been separated. In our example, integration was applied to both sides of the equation \( \frac{dy}{a + y} = \cot x \, dx \).
The left side, \( \int \frac{dy}{a+y} \), simplifies through a natural logarithm to \( \ln |a+y| \). Understanding basic integration techniques like these is fundamental when solving differential equations. For the right side, integration of \( \cot x \) required knowledge that \( \int \cot x \, dx = \ln |\sin x| \), a common integration in trigonometry. Completing these integrals allowed forming the solution equation \( \ln |a+y| = \ln |\sin x| + C \). This equation was then solved for \( y \) after exponentiation and using initial conditions, demonstrating the importance of correct integration techniques.

Trigonometric functions are often encountered in differential equations, especially in those involving periodic or oscillatory systems. In this exercise, the use of trigonometric identities and integrations played a key role.
The original equation involved \( \tan x \), which was transformed into \( \cot x \) using the identity \( \frac{1}{\tan x} = \cot x \). This manipulation is crucial for separating variables correctly and integrating them. When integrating \( \cot x \), understanding it yields \( \ln |\sin x| \) is vital. Concepts of trigonometry support both transformations and integrations, with \( \sin \), \( \cos \), \( \tan \), and \( \cot \) functions commonly employed. By handling these functions effectively, you gain a greater ability to solve intricate differential problems accurately.