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
Then
is given by
which is a constant. Therefore, the equation becomes a second-order ordinary differential equation with constant coefficients
 |
(15)
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Define
and
The solutions are
![y={c_1e^(r_1z)+c_2e^(r_2z) (alpha-1)^2>4beta; (c_1+c_2z)e^(az) (alpha-1)^2=4beta; e^(az)[c_1cos(bz)+c_2sin(bz)] (alpha-1)^2<4beta.](http://mathworld.wolfram.com/images/equations/EulerDifferentialEquation/NumberedEquation9.gif) |
(22)
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In terms of the original variable
,
![y={c_1|x|^(r_1)+c_2|x|^(r_2) (alpha-1)^2>4beta; (c_1+c_2ln|x|)|x|^a (alpha-1)^2=4beta; |x|^a[c_1cos(bln|x|)+c_2sin(bln|x|)] (alpha-1)^2<4beta.](http://mathworld.wolfram.com/images/equations/EulerDifferentialEquation/NumberedEquation10.gif) |
(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.