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Exact Solutions > Ordinary Differential Equations > Second-Order Linear Ordinary Differential Equations >Modified Bessel Equation

14. x2y′′xx + xy′

x – (x2 + ν2)y = 0.

Modified Bessel equation.It can be reduced to theBessel equationby means of the substitutionx = ix, wherei2 = −1.

Solution:y = C1Iν(x) + C2Kν(x),

whereC1 andC2 are arbitrary constants,Iν(x) andKν(x) are the modified Bessel functions of thefirst and second kind:

Iν(x) =∞∑

k=0

(x/2)2k+ν

k! Γ(ν + k + 1), Kν(x) =

π

2I−ν(x) − Iν(x)

sinπν.

ReferencesBateman, H. and Erdelyi, A., Higher Transcendental Functions, Vol. 2, McGraw-Hill, New York, 1953.Abramowitz, M. and Stegun, I. A. (Editors), Handbook of Mathematical Functions with Formulas, Graphs and

Mathematical Tables, National Bureau of Standards Applied Mathematics, Washington, 1964.Polyanin, A. D. and Zaitsev, V. F.,Handbook of Exact Solutions for Ordinary Differential Equations, 2nd Edition, Chapman

& Hall/CRC, Boca Raton, 2003.

Modified Bessel Equation

Copyright c© 2004 Andrei D. Polyanin http://eqworld.ipmnet.ru/en/solutions/ode/ode0214.pdf