91Ó°ÊÓ

Integration is a fundamental concept in calculus, particularly useful in solving differential equations. In many problems, including the one in our original exercise, integrating functions helps us find the general solution for a variable of interest. When we talk about integration techniques, we refer to various methods used to solve integrals, which can differ in complexity and application.

Common integration techniques include:

• Recognizing standard integrals - knowing the antiderivatives of common functions like polynomials, exponentials, and trigonometric functions.
• Substitution method - useful when the integral can be simplified by a change of variables.
• Integration by parts - based on the product rule for differentiation and used for products of functions.

In the context of our exercise, we apply integration to both sides of the separated differential equation, using the standard antiderivatives:

• Integrate the left side with respect to y, resulting in \( \int y \, dy = \frac{1}{2}y^2 + C_1 \).
• On the right, integrate the function \( e^{-x} \), yielding \( \int e^{-x} \, dx = -e^{-x} + C_2 \).

These integrals are straightforward because they involve simple exponential and polynomial forms, which underscores the accessibility of standard integration techniques.

Differential equations are equations that involve unknown functions and their derivatives. They come in various forms and are vital in modeling real-world phenomena where quantities change over time or space, such as physics, biology, and engineering.

There are different types of differential equations, which include:

• Ordinary differential equations (ODEs) - involve one independent variable and its derivatives.
• Partial differential equations (PDEs) - involve multiple independent variables.
• Linear vs. non-linear - based on whether the function and its derivatives appear linearly.

In our problem, we deal with a first-order ordinary differential equation (ODE), expressed as \( \frac{\mathrm{d} y}{\mathrm{d} x}=\frac{\mathrm{e}^{-x}}{y} \). This equation is 'separable', which means we can manipulate it to isolate the differentials and integrate both sides easily. Solving such equations gives us insights into how the dependent variable, in this case, y, changes with respect to the independent variable x.

Separation of variables is a method for solving differential equations where variables can be rearranged so that each side of the equation contains only one function of a single variable. This method is particularly effective for solving first-order separable differential equations.

To apply separation of variables, follow these steps:

• Identify if the equation is separable - Check if the equation can be rewritten as a product of two functions, one solely depending on y and the other on x.
• Rearrange the equation - Organize the terms such that all y-related terms (including dy) are on one side and all x-related terms (including dx) are on the other.
• Integrate both sides separately - Use integration techniques to solve for the general solution.

In our exercise, the separation of variables starts by rearranging the original equation \( \frac{\mathrm{d} y}{\mathrm{d} x}=\frac{\mathrm{e}^{-x}}{y} \) to \( y \frac{dy}{dx} = e^{-x} \), and then further to \( y \, dy = e^{-x} \, dx \). By doing this, we enable the integration process to proceed smoothly, ultimately allowing us to find the function y in terms of x. This method is not only systematic but also quite elegant in solving such types of equations.