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Removal of the first derivative from a second-order differential equation

For

y^{''} + P (x) y^{'} + Q (x) y = 0,

the substitution

y = e^{- \frac{1}{2} \int P (x) d x} v

gives the normal form

v^{''} + (Q - \frac{1}{2} P^{'} - \frac{1}{4} P^{2}) v = 0.

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