Problem 52 - Advanced Engineering Mathematics

Question

(a) The Fresnel sine integral is defined by \(S(x)=\) \(\int_{0}^{x} \sin \left(\pi t^{2} / 2\right) d t\). Express the solution \(y(x)\) of the initialvalue problem \(y^{\prime}-\left(\sin x^{2}\right) y=0, y(0)=5\), in terms of \(S(x)\) (b) Use a CAS to graph the solution curve for the IVP on \((-\infty, \infty)\) (c) It is known that \(S(x) \rightarrow \frac{1}{2}\) as \(x \rightarrow \infty\) and \(S(x) \rightarrow-\frac{1}{2}\) as \(x \rightarrow-\infty\). What does the solution \(y(x)\) approach as \(x \rightarrow \infty ?\) As \(x \rightarrow-\infty ?\) (d) Use a CAS to find the values of the absolute maximum and the absolute minimum of the solution \(y(x)\).

Step-by-Step Solution

Answer

(a) The solution is \( y(x)=5e^{-S(x)} \). (b) Graph using CAS. (c) Asymptotic values are \( 5e^{-1/2} \) and \( 5e^{1/2} \). (d) Find extrema using CAS.

Step 1: Understand the Differential Equation

The given differential equation is a first-order linear differential equation of the form \( y' - p(x) y = 0 \), with \( p(x) = \sin(x^2) \).

Step 2: Find the Integrating Factor

For a linear differential equation \( y' - p(x) y = 0 \), the integrating factor is \( e^{\int p(x) \, dx} \). Substitute \( p(x) = \sin(x^2) \) and find the integrating factor: \( e^{\int \sin(x^2) \, dx} = e^{S(x)} \).

Step 3: Solve the Differential Equation

Using the integrating factor \( e^{S(x)} \), multiply through the differential equation to obtain \( e^{S(x)} y' = 0 \). Integrate both sides to find \( e^{S(x)} y = C \). Solve for \( y \) to obtain \( y(x) = Ce^{-S(x)} \).

Step 4: Apply Initial Condition

Use the initial condition \( y(0) = 5 \) to find the constant \( C \). Since \( S(0) = 0 \), it follows that \( y(0) = Ce^{0} = 5 \), thus \( C = 5 \). Therefore, the solution is \( y(x) = 5e^{-S(x)} \).

Step 5: Graph the Solution

Use a CAS (Computer Algebra System) to graph the function \( y(x) = 5e^{-S(x)} \) over the interval \((-fty, fty)\). This will reveal the behavior of the solution curve across the given domain.

Step 6: Analyze Asymptotic Behavior

Given that \( S(x) \rightarrow \frac{1}{2} \) as \( x \rightarrow \infty \) and \( S(x) \rightarrow -\frac{1}{2} \) as \( x \rightarrow -\infty \), find \( y(x) \). As \( x \rightarrow \infty \), \( y(x) \rightarrow 5e^{-1/2} \). As \( x \rightarrow -\infty \), \( y(x) \rightarrow 5e^{1/2} \).

Step 7: Find Absolute Extrema

Using a CAS, find the points where \( y(x) = 5e^{-S(x)} \) reaches its maximum and minimum overall values. This involves evaluating \( y(x) \) at critical points as well as at asymptotic limits.

Key Concepts

Fresnel Sine Integral

The Fresnel Sine Integral is a special function denoted by \( S(x) \). It is defined by the integral:\[S(x) = \int_{0}^{x} \sin \left( \frac{\pi t^2}{2} \right) \, dt\]This function arises in various physics and engineering applications, particularly in optics and wave propagation.
This peculiar integral doesn't have a simple closed-form expression. Still, it can be evaluated using numerical methods or specialized software. Some key features of the Fresnel Sine Integral include:

• As \( x \) approaches positive or negative infinity, \( S(x) \) approaches specific constants, which are important for analyzing the behavior of solutions in related differential equations.
• The function is an odd function, meaning \( S(-x) = -S(x) \).

Initial Value Problem

An Initial Value Problem (IVP) in differential equations is a common scenario where the goal is to find a function that satisfies a given differential equation and meets an initial condition at a specific point.
In the exercise, the differential equation is:\[y' - \sin(x^2)y = 0\]with the initial condition given as \( y(0) = 5 \). This IVP requires us to find the function \( y(x) \) that satisfies both the differential equation and the initial condition.Key points to understand about IVPs:

• The initial condition provides a specific value for the solution at a particular point, ensuring a unique solution (under suitable conditions for the differential equation).
• Solving an IVP typically involves using integration techniques and applying the given conditions to determine constants of integration.

Integrating Factor

The integrating factor method is a technique used for solving linear first-order differential equations.
This technique involves multiplying the entire differential equation by a specific function, known as the integrating factor, to simplify the equation.The formula for the integrating factor in an equation of the form \( y' - p(x) y = 0 \) is:\[e^{\int p(x) \, dx}\]For this exercise, where \( p(x) = \sin(x^2) \), the integrating factor becomes \( e^{S(x)} \).
This enables us to express the differential equation in a form that can be directly integrated.Benefits of using an integrating factor include:

• Converts a non-exact differential equation into an exact one, allowing straightforward integration.
• The method is systematic, providing a step-by-step procedure applicable to a wide range of linear differential equations.

Asymptotic Behavior

Asymptotic behavior refers to the tendency of a function as the variable approaches a particular limit, often infinity or negative infinity. It's a way to describe what the function looks like at the extremes.
For the function \( S(x) \) in the Fresnel Sine Integral, the known results are:

• \( S(x) \rightarrow \frac{1}{2} \) as \( x \rightarrow \infty \)
• \( S(x) \rightarrow -\frac{1}{2} \) as \( x \rightarrow -\infty \)

This affects the solution \( y(x) = 5e^{-S(x)} \) of the differential equation:

• As \( x \rightarrow \infty \), \( y(x) \) tends toward \( 5e^{-1/2} \).
• As \( x \rightarrow -\infty \), \( y(x) \) trends toward \( 5e^{1/2} \).

Understanding asymptotic behavior is crucial as it provides insight into the long-term behavior of solutions, which is often more important than the specific values over a finite interval.