91Ó°ÊÓ

A differential equation is a mathematical equation that relates a function with its derivatives. In practice, it models how a certain quantity changes over time or space. The equation provided in the exercise, \( \frac{d y}{d x} = \frac{x}{y} \), is an example of a first-order differential equation. It links the derivative of the function \( y \) with respect to \( x \) to a ratio involving \( x \) and \( y \).

Understanding how to solve this type of equation is crucial for describing various physical phenomena, such as growth rates in biology, motion in physics, or changing rates in economics. The solution process often requires manipulating the equation to isolate the derivative on one side. In this case, through separation of variables, the equation was manipulated into a form where \( y \) and \( x \) are on opposite sides, facilitating the integration process that leads to the general solution.

Integrating factors are a technique used to solve certain differential equations that cannot be easily separated. While the equation from the exercise, \( \frac{d y}{d x} = \frac{x}{y} \), is solveable via separation of variables, not all differential equations are this straightforward. An integrating factor is a function that is multiplied by a given differential equation, making it possible to rewrite the equation as the derivative of a product of functions.

In general, for a linear first-order differential equation in the form \( \frac{d y}{d x} + P(x) y = Q(x) \), the integrating factor can be found by calculating the exponentiated integral of \( P(x) \) and can transform the equation into an exact equation. Although the integrating factor method was not required for the original equation, it’s a valuable strategy in a differential equations toolkit, especially for equations that do not separate naturally.

Finally, the general solution of a differential equation is an expression that contains all possible solutions to the equation. It is called 'general' because it includes a constant of integration \( C \), which can take on any value. In our exercise, after integrating both sides of the separated equation, we arrive at \(0.5 y^2 = 0.5 x^2 + C\). Simplifying this, we achieve the general solution \( y = \pm \sqrt{x^2 + 2C} \).

The \( \pm \) sign in front of the square root acknowledges that the function \( y \) could be either positive or negative, allowing the solution to cover all possible cases of the square of a number. The constant \( C \) represents the indefinite integral's arbitrary constant, reflecting the infinite set of curves that satisfy the equation, each curve corresponding to a different initial condition or starting point. The general solution captures the essence of all these infinitely many specific solutions.