91Ó°ÊÓ

Differential equations are equations that involve an unknown function and its derivatives. They play a crucial role in both pure and applied mathematics. For example, the given differential equation in the problem is: \[ y' = x \, \sqrt{x^2 + 1} \]. Here, \( y' \), or \( \frac{dy}{dx} \), represents the derivative of \( y \) with respect to \( x \). Solving this differential equation means finding the function \( y(x) \) that satisfies this equation for all \( x \). This function will tell us how \( y \) changes as \( x \) changes. Hence, differential equations can model many phenomena from physics, engineering, and other sciences.

In this exercise, integration is used to solve a differential equation. Integration is essentially the reverse process of differentiation. It helps to find the original function given its derivative. When we rewrite the given differential equation as \( \frac{dy}{dx} = x \sqrt{x^2 + 1} \), integrating both sides with respect to \( x \) leads to: \[ \int dy = \int x \sqrt{x^2 + 1} \, dx \]. The left side is straightforward, resulting in \( y \), while the right side requires more work. This process involves identifying an appropriate substitution to simplify the integral.

The substitution method is a powerful technique for solving integrals that are otherwise difficult to handle. In this problem, we use substitution to simplify the integral on the right side of the equation. By letting \( u = x^2 + 1 \), we transform the integral into a more manageable form: \[ du = 2x \, dx \]. This allows us to rewrite the integral as: \[ \int x \sqrt{x^2 + 1} \, dx = \frac{1}{2} \int u^{1/2} \, du \]. Integrating \( u^{1/2} \) is straightforward and gives us: \[ \frac{1}{2} \cdot \frac{2}{3} u^{3/2} = \frac{1}{3} u^{3/2} = \frac{1}{3} (x^2 + 1)^{3/2} \]. By reverting back to \( x \) from \( u \), we solve the original integral.

Initial conditions are specific values given for the function or its derivatives at a particular point. They allow us to find a unique solution to a differential equation, known as the particular solution. In this problem, the initial condition is \( y(0) = 3 \). After finding the general solution: \[ y = \frac{1}{3} (x^2 + 1)^{3/2} + C \], we substitute \( x = 0 \) and \( y = 3 \) to determine the constant \( C \): \[ 3 = \frac{1}{3} (0^2 + 1)^{3/2} + C \]. Simplifying this, we get \[ 3 = \frac{1}{3} + C \], which gives \( C = \frac{8}{3} \). Thus, the particular solution is: \[ y = \frac{1}{3} (x^2 + 1)^{3/2} + \frac{8}{3} \]. Initial conditions ensure that the solution fits the specific scenario described by the problem.