91Ó°ÊÓ

Understanding antiderivatives is a crucial step in solving differential equations like the ones given in the exercise. In simple terms, an antiderivative is the reverse process of differentiation. It involves finding a function whose derivative is the original function you started with. In the context of this exercise:

• The antiderivative of \( rac{dy}{dx} = ext{sin}3x \) is \( y = -\frac{1}{3} ext{cos}3x + C \).
• Similarly, for the equation \( rac{dy}{dx} = ext{cos}3x \), the antiderivative is \( y = \frac{1}{3} ext{sin}3x + C \).

This process involves applying basic integration rules. When integrating trigonometric functions like \( ext{sin} \) and \( ext{cos} \), we essentially look for the general formula that translates back to the derivative we started with. This requires us to remember that:

• The derivative of \( ext{sin}x \) is \( ext{cos}x \).
• The derivative of \( ext{cos}x \) is \(- ext{sin}x \).

Recognizing these relationships is key in calculating the antiderivatives correctly. Once these antiderivatives are determined, the solution is expressed by including an arbitrary constant \( C \), representing the infinite number of possible functions that all share the same derivative.

First-order differential equations are those where the highest derivative, or rate of change, is the first derivative. They're crucial in modeling a wide array of real-world phenomena, from physical systems to population dynamics. In our context, the differential equations \( \frac{dy}{dx} = \sin 3x \) and \( \frac{dy}{dx} = \cos 3x \) are prime examples of such equations. These equations are classified as **separable**, meaning they can be rewritten to allow the variables \( x \) and \( y \) to be on different sides of the equation. This property simplifies integration since each variable can be independently integrated. Here's what makes these equations distinct:

• **Separable:** the ability to express the equation such that one side contains only derivatives related to \( y \), and the other side involves only \( x \).
• **Ordinary:** only involves one independent variable, in this case, \( x \).
• **Straightforward to solve:** through integration techniques when separable, the solution is often reachable with basic calculus knowledge.

This approach makes it easier for students to build a bridge between recognizing a differential form and using integral methods to find solutions.

When solving differential equations, each antiderivative obtained will have a constant added to it - the constant of integration, denoted as \( C \). This constant represents the family of all possible solutions that differ by a constant value, indicating the dependency of the solution on initial conditions or boundary conditions, if known. Consider the two solutions from the exercise:

• \( y = -\frac{1}{3}\text{cos} 3x + C \)
• \( y = \frac{1}{3}\text{sin} 3x + C \)

The presence of \( C \) accounts for an infinite set of functions, all valid solutions which could describe different physical scenarios. Each value of \( C \) would result in a different function that closely fits specific situations. Therefore, if initial conditions are provided, they can be plugged into the general solution to solve for the exact value of \( C \), making the solution specific to a particular problem. Understanding the constant of integration's role is pivotal in controlling solutions' specificity and predicting behaviors accurately in mathematical modeling.