The general nonhomogeneous equation is
![\begin{displaymath}
x^2 {d^2y\over dx^2} + \alpha x {dy\over dx} + \beta y = S(x).
\end{displaymath}](e_2056.gif) |
(1) |
The homogeneous equation is
![\begin{displaymath}
x^2y''+\alpha xy'+\beta y = 0
\end{displaymath}](e_2057.gif) |
(2) |
![\begin{displaymath}
y'' + {\alpha\over x} y' + {\beta\over x^2} y = 0.
\end{displaymath}](e_2058.gif) |
(3) |
Now attempt to convert the equation from
![\begin{displaymath}
y''+p(x)y'+q(x)y=0
\end{displaymath}](e_2059.gif) |
(4) |
to one with constant Coefficients
![\begin{displaymath}
{d^2y\over dz^2} + A {dy\over dz} + By = 0
\end{displaymath}](e_2060.gif) |
(5) |
by using the standard transformation for linear Second-Order Ordinary Differential Equations. Comparing (3) and (5), the functions
and
are
![\begin{displaymath}
p(x) \equiv {\alpha\over x} = \alpha x^{-1}
\end{displaymath}](e_2062.gif) |
(6) |
![\begin{displaymath}
q(x) \equiv {\beta\over x^2} = \beta x^{-2}.
\end{displaymath}](e_2063.gif) |
(7) |
Let
and define
Then
is given by
which is a constant. Therefore, the equation becomes a second-order ODE with constant Coefficients
![\begin{displaymath}
{d^2y\over dz^2}+(\alpha-1){dy\over dz}+\beta y = 0.
\end{displaymath}](e_2070.gif) |
(10) |
Define
and
The solutions are
In terms of the original variable
,
© 1996-9 Eric W. Weisstein
1999-05-25