Problem 51 - Advanced Engineering Mathematics

Question

(a) The sine integral function is defined by \(\operatorname{Si}(x)=\) \(\int_{0}^{x}(\sin t / t) d t\), where the integrand is defined to be 1 at \(t=0\). Express the solution \(y(x)\) of the initial-value problem \(x^{3} y^{\prime}+2 x^{2} y=10 \sin x, y(1)=0\), in terms of \(\operatorname{Si}(x)\). (b) Use a CAS to graph the solution curve for the IVP for \(x>0 .\) (c) Use a CAS to find the value of the absolute maximum of the solution \(y(x)\) for \(x>0\).

Step-by-Step Solution

Answer

The solution is \( y(x) = \frac{10}{x^2} (\operatorname{Si}(x) - \operatorname{Si}(1)) \). Use CAS to graph and find maxima.

Step 1: Understanding the Differential Equation

The given initial value problem is a first-order linear differential equation: \(x^{3} y^{\prime} + 2 x^{2} y = 10 \sin(x)\), with the initial condition \(y(1) = 0\). Our task is to find the solution \(y(x)\) in terms of the sine integral function \(\operatorname{Si}(x)\).

Step 2: Rewriting the Equation

Rearrange the differential equation as \( y' + \frac{2}{x} y = \frac{10 \sin(x)}{x^3} \). This is now in the standard form to which we can apply the integrating factor method.

Step 3: Finding the Integrating Factor

The integrating factor \(\mu(x)\) is found using the formula \(\mu(x) = e^{\int \frac{2}{x} dx}\). This simplifies to \(\mu(x) = x^2\).

Step 4: Solving the Differential Equation

Multiply the entire differential equation by the integrating factor \(x^2\) to get \(x^2 y' + 2xy = \frac{10 \sin(x)}{x}\). The left side is now the derivative of \(x^2 y\) with respect to \(x\). Integrate both sides with respect to \(x\): \[x^2 y = \int \frac{10 \sin(x)}{x} dx.\] The right side involves the sine integral function, thus: \[x^2 y = 10 \operatorname{Si}(x) + C.\]

Step 5: Applying the Initial Condition

Use \(y(1) = 0\) to solve for the constant \(C\). Substituting \(x = 1\) and \(y(1) = 0\):\[1^2 \cdot 0 = 10 \operatorname{Si}(1) + C.\]Therefore, \(C = -10 \operatorname{Si}(1)\).

Step 6: Expression for Solution y(x)

Substitute \(C\) back into the equation:\[x^2 y = 10 \operatorname{Si}(x) - 10 \operatorname{Si}(1),\]thus:\[y(x) = \frac{10}{x^2} (\operatorname{Si}(x) - \operatorname{Si}(1)).\]

Step 7: Graphing the Solution Curve

To graph the solution curve \(y(x)\) for \(x>0\), use a Computer Algebra System (CAS) to plot the function \(y(x) = \frac{10}{x^2} (\operatorname{Si}(x) - \operatorname{Si}(1))\).

Step 8: Finding the Absolute Maximum

Use the CAS to compute the derivative of \(y(x)\) and find critical points. Evaluate \(y(x)\) at these points to determine the absolute maximum value for \(x>0\).

Key Concepts

First-order Linear Differential Equations

First-order linear differential equations are equations that can be written in the form \( y' + P(x) y = Q(x) \), where \( y' \) is the derivative of \( y \) with respect to \( x \), and \( P(x) \) and \( Q(x) \) are continuous functions on an interval. These equations arise frequently in mathematical modeling and have wide applications across science and engineering. They are called 'linear' because they only involve the first power of \( y \) and its derivative, without any products or powers.
To solve a first-order linear differential equation, the goal is usually to find an expression for \( y \) in terms of \( x \) and possibly some other known functions. The equation provides a mathematical relationship involving \( y \) and its rate of change, allowing one to study how \( y \) evolves with \( x \). In the exercise, the given equation falls into this category, transforming into a more familiar form after some rearrangements, enabling further analysis through consistent methodologies like the integrating factor method.

Initial Value Problem

An initial value problem (IVP) in the context of differential equations specifies not just the differential equation itself, but also an initial condition. This condition is typically expressed as \( y(x_0) = y_0 \), where \( x_0 \) is a point in the domain of the solution, and \( y_0 \) is the specified value of the solution at that point.
The initial condition helps to determine a unique solution curve among potentially infinite solutions of the differential equation. By establishing a specific "starting point" with \( y(1) = 0 \) in the original problem, the solution is tailored to satisfy both the equation and this specific value, providing a complete picture of the behavior of the variable \( y \) as a function of \( x \). Without this initial condition, the solution to the differential equation might contain arbitrary constants, leading to many possible solutions rather than a single definitive path.

Integrating Factor Method

The integrating factor method is a systematic technique for solving first-order linear differential equations. It centers around finding a function, known as the integrating factor \( \mu(x) \), that transforms a differential equation into a form that is easier to solve.
The procedure involves determining \( \mu(x) = e^{\int P(x) dx} \), meaning the exponential of the integral of the function \( P(x) \), which appears in the standard form of the differential equation. Once the integrating factor is identified, the original equation is modified by multiplying through by \( \mu(x) \). This folds the left-hand side into the derivative of the product \( \mu(x)y \), allowing for straightforward integration across the equation.
In the problem at hand, this approach facilitated the expression of the solution involving the sine integral function \( \operatorname{Si}(x) \). By integrating the modified equation, the involved calculus helps link it to known mathematical functions, making the solution more elegant and easier to interpret.

Computer Algebra System (CAS)

A Computer Algebra System (CAS) is an invaluable tool in solving and visualizing complex mathematical problems. These systems provide capabilities for algebraic manipulation, calculus operations, graphing, and even solving equations analytically. Typical applications include simplifying or factoring expressions, computing symbolic derivatives and integrals, numerical evaluations, and visualizing functions.
In the context of the exercise, a CAS plays two pivotal roles:

• First, it is used to graph the solution curve \( y(x) = \frac{10}{x^2} (\operatorname{Si}(x) - \operatorname{Si}(1)) \) for \( x > 0 \), offering a visual perspective of how the solution behaves over different values of \( x \).
• Second, the CAS assists in determining critical points to find the absolute maximum of the solution. By differentiating \( y(x) \), setting the derivative to zero, and solving for \( x \), it provides insights into where the function achieves its highest value in the specified domain.

Modern CAS tools like Mathematica, Maple, or online alternatives such as Wolfram Alpha are excellent resources for both educational and research purposes, enabling users to make precise and efficient calculations.