Question

{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.

Accepted Answer

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. 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. 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. 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.$$  
Multiply each derivative by the appropriate powers of x$ as indicated, and combine like terms by powers of x$ up to x^6$. Verify that the sum of all terms up to x^6$ cancels out to zero, confirming that the truncated series satisfies the differential equation approximately. This demonstrates that J_0$(x) is a $$solution to the Bessel differential equation up to the considered order.

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Key Concepts

Power Series Representation

Differentiation of Power Series

Bessel's Differential Equation