Reciprocal-coordinate reduction of a beam-type equation
= Reciprocal-coordinate reduction of a beam-type equation
{title2=$u=t[A\cos(\lambda/t)+B\sin(\lambda/t)]$}
For $t>0$, the <change of variables> $\tau=1/t$ and $u(t)=v(\tau)/\tau$ converts $t^4u''+\lambda^2u=0$ into $v''+\lambda^2v=0$. The <chain rule> gives $u'=v-\tau v'$ and $u''=\tau^3v''$. Thus $u=t[A\cos(\lambda/t)+B\sin(\lambda/t)]$. The prefactor cancels the derivative term introduced by inverting the independent variable.