 Solution

With $\tau =1/t$, the chain rule gives $d/dt=-{\tau }^{2}d/d\tau$. Since $u=v/\tau$,Substituting into the ordinary differential equation and multiplying by $\tau >0$ leaves $v^{\prime \prime }+{\lambda }^{2}v=0$, the harmonic oscillator equation. Its two independent real solutions are $\cos (\lambda \tau )$ and $\sin (\lambda \tau )$. Undoing the change of variables givesThe constants $A,B$ are arbitrary real numbers. The reciprocal-coordinate reduction of a beam-type equation works on the entire positive half-line, where the coordinate transformation is invertible.