Problem 27 - Calculus

Question

Find solutions to \(d y / d x=\sin 3 x\) and \(d y / d x=\cos 3 x\).

Step-by-Step Solution

Answer

Solutions: \(y = -\frac{1}{3}\cos 3x + C\) and \(y = \frac{1}{3}\sin 3x + C\).

Step 1: Identify Differential Equation Type for Equation 1

The first equation given is a separable first-order ordinary differential equation: \(\frac{dy}{dx} = \sin 3x\). This implies that the solution is an antiderivative problem.

Step 2: Integrate Equation 1

Integrate \(\sin 3x\) with respect to \(x\) to find \(y\). The antiderivative of \(\sin 3x\) is \(-\frac{1}{3}\cos 3x\) since the derivative of \(-\cos 3x\) is \(3\sin 3x\), thus when divided by 3, it matches our integrand. The general solution is \(y = -\frac{1}{3}\cos 3x + C\) where \(C\) is the constant of integration.

Step 3: Identify Differential Equation Type for Equation 2

The second equation is also a separable first-order ordinary differential equation: \(\frac{dy}{dx} = \cos 3x\), indicating another antiderivative problem.

Step 4: Integrate Equation 2

Integrate \(\cos 3x\) with respect to \(x\) to find \(y\). The antiderivative of \(\cos 3x\) is \(\frac{1}{3}\sin 3x\) since the derivative of \(\sin 3x\) is \(3\cos 3x\), thus when divided by 3, it matches our integrand. The general solution is \(y = \frac{1}{3}\sin 3x + C\) where \(C\) is the constant of integration.

Key Concepts

Antiderivatives

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

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.

Constant of Integration

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.