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Modified Bessel Differential Equation

The second-order ordinary differential equation

x^2(d^2y)/(dx^2)+x(dy)/(dx)-(x^2+n^2)y=0.
(1)

The solutions are the modified Bessel functions of the first and second kinds, and can be written

y=a_1J_n(-ix)+a_2Y_n(-ix)
(2)
=c_1I_n(x)+c_2K_n(x),
(3)

where J_n(x) is a Bessel function of the first kind, Y_n(x) is a Bessel function of the second kind, I_n(x) is a modified Bessel function of the first kind, and K_n(x) is modified Bessel function of the second kind.

If n=0, the modified Bessel differential equation becomes

x^2(d^2y)/(dx^2)+x(dy)/(dx)-x^2y=0,
(4)

which can also be written

d/(dx)(x(dy)/(dx))=xy.
(5)

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References

Abramowitz, M. and Stegun, I. A. (Eds.). §9.6.1 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, 1972.Zwillinger, D. Handbook of Differential Equations, 3rd ed. Boston, MA: Academic Press, p. 121, 1997.

Referenced on Wolfram|Alpha

Modified Bessel Differential Equation

Cite this as:

Weisstein, Eric W. "Modified Bessel Differential Equation." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/ModifiedBesselDifferentialEquation.html

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