Problem 29 - Advanced Engineering Mathematics

Question

$$ \int_{0}^{x} r J_{0}(r) d r=x J_{1}(x) $$

Step-by-Step Solution

Answer

The integral is equal to \( x J_{1}(x) \).

Step 1: Understanding the Problem

We need to evaluate the definite integral \( \int_{0}^{x} r J_{0}(r) \, dr \) and show that it is equal to \( x J_{1}(x) \), where \( J_{0} \) and \( J_{1} \) are Bessel functions of the first kind of order 0 and 1, respectively.

Step 2: Recall Bessel Function Properties

The Bessel functions \( J_{0}(x) \) and \( J_{1}(x) \) are solutions to Bessel's differential equation. An important identity for the Bessel functions of the first kind is: \( \frac{d}{dx}[xJ_{1}(x)] = xJ_{0}(x) \).

Step 3: Differentiate the Right Side

Differentiate the right-hand expression \( x J_{1}(x) \) with respect to \( x \):\[ \frac{d}{dx}(xJ_{1}(x)) = J_{1}(x) + xJ_{1}'(x) \].Using the identity \( J_{1}'(x) = -J_{0}(x) \), this simplifies to \( J_{1}(x) + x(-J_{0}(x)) = J_{1}(x) - xJ_{0}(x) \).

Step 4: Apply Fundamental Theorem of Calculus

To solve the integral \( \int_{0}^{x} r J_{0}(r) \, dr \), recognize it as a result of the anti-derivative. From Step 3, since \( \frac{d}{dx}(xJ_{1}(x)) = xJ_{0}(x) \), by the Fundamental Theorem of Calculus:\[ \int_{0}^{x} r J_{0}(r) \, dr = xJ_{1}(x) - 0 \cdot J_{1}(0) = xJ_{1}(x) \].

Step 5: Verify the Boundary Conditions

Evaluate the boundary conditions of this integral. At \( x = 0 \), both sides are zero since \( x J_{1}(x) \) equals zero and the definite integral from 0 to 0 equals zero. Thus, the equality holds for all \( x \geq 0 \).

Key Concepts

Definite Integral

A definite integral is a mathematical expression that represents the area under the curve of a function, between two specific limits. In our original exercise, the definite integral is \[ \int_{0}^{x} r J_{0}(r) \, dr \] Here, the function we are integrating is \( r J_{0}(r) \), where \( J_{0} \) is a Bessel function of the first kind of order 0. This type of integral provides valuable information about how functions behave over intervals, particularly when they involve special functions like Bessel functions which arise frequently in physics and engineering.

• The lower limit of integration is 0, and the upper limit is \( x \), a variable which can take any non-negative value.
• The result of a definite integral is usually a number, which in this case, depends on \( x \).

This integral specifically has significant implications in solving differential equations where Bessel functions are involved.

Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus bridges the concept of differentiation with integration, two main concepts in calculus. It tells us that if a function has an anti-derivative, then the value of its definite integral can be calculated using this anti-derivative.
In simple terms, it states:

• The definite integral of a function from \( a \) to \( b \) is the difference in the values of the anti-derivative at \( b \) and at \( a \).

In the original problem, we used the Fundamental Theorem of Calculus to resolve the indefinite integral of the expression on the left. By recognizing that differentiating \( x J_{1}(x) \) gives us \( x J_{0}(x) \), we can say that the integral of \( x J_{0}(x) \) from 0 to \( x \) must equal \( x J_{1}(x) \). This powerful theorem makes complex integration directly manageable when the anti-derivative is known or calculated.

Bessel's Differential Equation

Bessel's differential equation arises in problems with cylindrical symmetry, common in physics and engineering fields like acoustics, electromagnetism, and fluid dynamics. The equation is \[x^2 y'' + x y' + (x^2 - n^2) y = 0\] where \( n \) is a constant. Bessel functions, including \( J_{0}(x) \) and \( J_{1}(x) \), are solutions to this differential equation.
Key aspects of Bessel functions include:

• They are oscillatory functions that generalize trigonometric functions but are suited for radial components.
• \( J_{0}(x) \) represents the zeroth-order Bessel function, while \( J_{1}(x) \) represents the first order. Both are routinely encountered in wave propagation and heat conduction problems.

In the exercise, we verified that specific properties of these functions coincide with known solutions to integrals involving these equations. This nature of Bessel functions helps in solving and simplifying real-world physical problems effectively.