The general nonhomogeneous differential equation is given by
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(1)
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and the homogeneous equation is
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(2)
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(3)
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Now attempt to convert the equation from
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(4)
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to one with constant coefficients
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(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
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(6)
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(7)
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Let and define
Then is given by
which is a constant. Therefore, the equation becomes a second-order ordinary differential equation with constant coefficients
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(15)
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Define
and
The solutions are
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(22)
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In terms of the original variable ,
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(23)
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Zwillinger (1997, p. 120) gives two other types of equations known as Euler differential equations,
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(24)
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(Valiron 1950, p. 201) and
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(25)
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(Valiron 1950, p. 212), the latter of which can be solved in terms of Bessel functions.