Elimination of the first derivative
= Elimination of the first derivative
For a <second-order linear differential equation>
$$
y''+p(x)y'+q(x)y=0,
$$
the substitution $y=u\exp[-\tfrac12\int p(x)\,dx]$ eliminates the first-derivative term and gives
$$
u''+\left(q-\frac12p'-\frac14p^2\right)u=0.
$$