The general nonhomogeneous differential equation is given by
(1)
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and the homogeneous equation is
(2)
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(3)
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Now attempt to convert the equation from
(4)
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to one with constant coefficients
(5)
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by using the standard transformation for linear second-order ordinary differential equations. Comparing (3) and (5), the functions and are
(6)
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(7)
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Let and define
(8)
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(9)
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(10)
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(11)
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Then is given by
(12)
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(13)
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(14)
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which is a constant. Therefore, the equation becomes a second-order ordinary differential equation with constant coefficients
(15)
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Define
(16)
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(17)
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(18)
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(19)
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and
(20)
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(21)
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The solutions are
(22)
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In terms of the original variable ,
(23)
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Zwillinger (1997, p. 120) gives two other types of equations known as Euler differential equations,
(24)
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(Valiron 1950, p. 201) and
(25)
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(Valiron 1950, p. 212), the latter of which can be solved in terms of Bessel functions.