{Use of Tech} Bessel functions Bessel functions arise in the study of wave propagation in circular geometries (for example, waves on a circular drum head). They are conveniently defined as power series. One of an infinite family of Bessel functions is
J₀(x) = ∑ₖ₌₀∞ (−1)ᵏ/(2²ᵏ(k!)²) x²ᵏ
c. Differentiate J₀ twice and show (by keeping terms through x⁶) that J₀ satisfies the equation x² y′′(x) + xy′(x) + x²y(x)=0.

Verified step by step guidance

1. Start with the given power series for the Bessel function of order zero: \[J_0(x) = \sum_{k=0}^\infty \frac{(-1)^k}{2^{2k} (k!)^2} x^{2k}.\] Write out the terms explicitly up to the power \(x^6\) to work with a finite polynomial approximation.
2. Differentiate \(J_0(x)\) term-by-term to find the first derivative \(J_0'(x)\). Use the power rule for differentiation: \[\frac{d}{dx} x^{2k} = 2k x^{2k-1}.\] Keep terms up to the power \(x^5\) since differentiating reduces the power by one.
3. Differentiate \(J_0'(x)\) term-by-term again to find the second derivative \(J_0''(x)\). Apply the power rule once more: \[\frac{d}{dx} x^{m} = m x^{m-1}.\] Keep terms up to the power \(x^4\) because the second derivative reduces the power by two from the original series.
4. Substitute \(J_0(x)\), \(J_0'(x)\), and \(J_0''(x)\) into the differential equation: \[x^2 J_0''(x) + x J_0'(x) + x^2 J_0(x) = 0.\] Multiply each derivative by the appropriate powers of \(x\) as indicated, and combine like terms by powers of \(x\) up to \(x^6\).
5. Verify that the sum of all terms up to \(x^6\) cancels out to zero, confirming that the truncated series satisfies the differential equation approximately. This demonstrates that \(J_0(x)\) is a solution to the Bessel differential equation up to the considered order.

Key Concepts

Power Series Representation

A power series expresses a function as an infinite sum of terms involving powers of the variable. For Bessel functions, this series form allows term-by-term differentiation and approximation by truncating after a finite number of terms, which is essential for verifying differential equations approximately.

Differentiation of Power Series

Differentiating a power series involves differentiating each term individually, which is valid within the radius of convergence. This process helps find the first and second derivatives of J₀(x) by applying standard differentiation rules to each power term, enabling substitution into the differential equation.

Bessel's Differential Equation

Bessel functions satisfy a specific second-order linear differential equation, here given as x²y'' + xy' + x²y = 0. Showing that the differentiated series satisfies this equation confirms the function's validity as a solution, linking the power series form to the underlying physical and mathematical properties.