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Standard reduction of order procedure: suppose there is a second solution of the form 
, which has derivatives
Substitute these terms into the ODE:
and replacing
, we have an ODE linear in 
:
Divide both sides by
, giving
and noting that the left hand side is a derivative of a product, namely
![\dfrac{\mathrm d}{\mathrm dx}[wx]=0](https://tex.z-dn.net/?f=%5Cdfrac%7B%5Cmathrm%20d%7D%7B%5Cmathrm%20dx%7D%5Bwx%5D%3D0)
we can then integrate both sides to obtain
Solve for
:
Now
where the second term is already accounted for by
, which means 
, and the above is the general solution for the ODE.
Substitute these terms into the ODE:
and replacing
Divide both sides by
and noting that the left hand side is a derivative of a product, namely
we can then integrate both sides to obtain
Solve for
Now
where the second term is already accounted for by
