Problem 43 - Advanced Engineering Mathematics

Question

Given that \(y=\sin x\) is an explicit solution of the first-order differential equation \(d y / d x=\sqrt{1-y^{2}} .\) Find an interval \(I\) of definition. [Hint: \(I\) is not the interval \((-\infty, \infty) .]\)

Step-by-Step Solution

Answer

The interval of definition is \([0, 2\pi]\).

Step 1: Identify Function Properties

First, understand that the given function is the sine function, \(y = \sin x\), which oscillates between -1 and 1. Hence \(y\) must satisfy \(-1 \leq y \leq 1\).

Step 2: Apply the Constraint of the Differential Equation

The differential equation \( \frac{dy}{dx} = \sqrt{1-y^2} \) implies \(1 - y^2 \geq 0\), which restricts \(y\) to \(-1 \leq y \leq 1\) as \( \sqrt{1-y^2} \) must be real.

Step 3: Consider the Domain of \(y = \sin x\)

The sine function is periodic with period \(2\pi\), and \(y = \sin x\) can oscillate from -1 to 1 multiple times. Since both the sine function and the square root in the differential equation are real, we need to identify a single interval where the sine function makes exactly one complete oscillation, such as \([0, 2\pi]\).

Step 4: Evaluate \(\sin x\) and \(\sqrt{1-y^2}\)

As \(\sin x\) is continuous and differentiable over \([0, 2\pi]\), its derivative \(\cos x\) matches \(\sqrt{1 - \sin^2 x} = |\cos x|\) in this interval. This confirms \(\sin x\) satisfies the differential equation in a single oscillation cycle.

Step 5: Determine Valid Interval \(I\)

Thus, given that \(y = \sin x\) fulfills the differential equation \(\frac{dy}{dx} = \sqrt{1 - y^2}\) on a complete single oscillation, the interval of definition is \(I = [0, 2\pi]\).

Key Concepts

Sine Function

The sine function, denoted as \( y = \sin x \), is a fundamental trigonometric function that describes a smooth, periodic oscillation. It emerges in many areas of mathematics and physics due to its simple harmonic properties.
It oscillates between -1 and 1, creating a wave-like pattern that repeats every \( 2\pi \) radians. This characteristic makes it suitable for modeling phenomena like sound waves or light waves.

• Characteristics of Sine Function:
  • Periodic: It repeats its values in regular cycles of \( 2\pi \).
  • Range: The values of \( y = \sin x \) are limited between -1 and 1.
  • Oscillation: It continuously oscillates through a maximum of 1 and a minimum of -1.
• Graphical Representation: The graph of the sine function is a smooth and continuous wave, beginning at zero, rising to a peak, falling to a trough, and returning to zero.

Interval of Definition

An interval of definition determines the set of input values for which a function or solution is defined and exists. For the function \( y = \sin x \), it is crucial to determine the appropriate interval where the solution satisfies the given conditions.
In the context of differential equations, the interval of definition restricts the domain of \( x \) to ensure that solutions like \( y = \sin x \) do not exhibit behavior that could cause issues, such as discontinuities or undefined expressions.

• Considerations for Interval of Definition:
  • Periodicity: Since \( y = \sin x \) repeats every \( 2\pi \), we often choose intervals like \([0, 2\pi]\) or similar segments for analysis.
  • Mathematical Constraints: Ensure the differential equation remains valid within the chosen interval, meaning calculations like \( \sqrt{1 - y^2} \) remain within real values.
• Practical Applications: Examples in engineering and physics often require finite intervals to calculate harmonics and waveforms.

First-Order Differential Equation

A first-order differential equation involves the first derivative of the function, linking the rate of change of a variable to the variable itself.
The given equation \( \frac{dy}{dx} = \sqrt{1-y^2} \) is an example of a first-order differential equation. Here it explicitly connects the derivative of \( y \) with a function involving \( y \) itself.

• Characteristics of First-Order Differential Equations:
  • Simplicity: They often represent systems with single dependencies, making them easier to solve compared to higher-order equations.
  • Integration Methods: Typically solved using separation of variables, substitution, or integrating factors.
• Applications: Widely used in modeling growth decay problems, velocity and acceleration in physics, or electrical circuits.

Oscillation

Oscillation describes the repetitive variations, typically in time, of some measure around a central value or between two or more different states.
The sine function \( y = \sin x \) exemplifies oscillatory behavior, cycling between -1 and 1.

• Key Features of Oscillation:
  • Amplitude: The maximum deviation from the central value (here, from 0, the amplitude is 1).
  • Period: The duration of one full cycle, \( 2\pi \) for the sine function.
  • Frequency: The number of cycles completed in a unit time, it is the inverse of the period.
• Significance in Mathematics: Oscillations are central to the study of waves and harmonic motion, helping us model phenomena like tides, market cycles, or circuits in alternating current systems.
• Visualizing Oscillation: Graphs of functions like \( y = \sin x \) help visualize how oscillatory values rise and fall cyclically.