= Solution
With $\tau=1/t$, the <chain rule> gives $d/dt=-\tau^2d/d\tau$. Since $u=v/\tau$,
$$
\frac{du}{dt}=v-\tau v',\qquad \frac{d^2u}{dt^2}=\tau^3v''.
$$
Substituting into the <ordinary differential equation> and multiplying by $\tau>0$ leaves $v''+\lambda^2v=0$, the <harmonic oscillator equation>. Its two independent real solutions are $\cos(\lambda\tau)$ and $\sin(\lambda\tau)$. Undoing the <change of variables> gives
$$
\boxed{u(t)=t\left[A\cos\frac\lambda t+B\sin\frac\lambda t\right],\qquad t>0.}
$$
The 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.
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