91Ó°ÊÓ

A first-order linear differential equation is an equation that involves the derivative of the unknown function and the function itself. It is called "first-order" because the highest derivative present is the first derivative. These equations take the form: \( \frac{d y}{d x} + P(x) y = Q(x) \). In this equation, \( y \) is the function of \( x \) we want to solve for, \( \frac{d y}{d x} \) is the first derivative of \( y \) with respect to \( x \), \( P(x) \) is a function of \( x \), and \( Q(x) \) is another function of \( x \). Understanding how to recognize and solve these equations is crucial for many fields of science and engineering. By transforming or simplifying these equations, we can describe a variety of real-world phenomena, from the cooling of an object to the growth of a population.

The integrating factor is a crucial tool for solving first-order linear differential equations. To transform the equation into a form that is easier to solve, we use the integrating factor \( \mu(x) \), defined as \( \mu(x) = e^{\int P(x) \, dx} \).

• The idea is to multiply the entire differential equation by \( \mu(x) \) to facilitate integration.
• This trick allows us to rewrite the equation such that the left side becomes the derivative of a product of functions.
• This technique essentially transforms the problem into one of solving a simpler integral equation.

For example, in the exercise, we are given \( P(x) = 2x \), leading us to calculate the integrating factor as \( e^{x^2} \). When applied, it simplifies the differential equation into a format where direct integration becomes feasible.

Integration by substitution is a powerful method used when direct integration of a function is challenging. It's akin to the reverse of the chain rule for derivatives, where we substitute a part of the integrand with a single variable to simplify the integration process. Consider the integral \( \int x e^{x^2} \, dx \) from the problem solution.

Steps for Substitution

• Choose a substitution: Let \( u = x^2 \). Then the differential \( du = 2x \, dx \) implies that \( \, x \, dx = \frac{1}{2} du \).
• Substitute in the integral: The integral becomes \( \frac{1}{2} \int e^u \, du \).
• Integrate: Since the integral of \( e^u \) is itself, this becomes \( \frac{1}{2} e^u + C \).
• Back-substitute: Replace \( u \) with \( x^2 \) to obtain the result in terms of \( x \).

This method effectively reduces complex expressions to simpler forms, making integration manageable even for seemingly complicated functions.