= Decaying profile for a fourth-order dilation equation
{title2=$D^4v-\beta Dv=0,\quad v(1)=0,\quad v'(1)=1$}
For real $\beta$, the displayed <boundary value problem> with $v(\xi)\to0$ as $\xi\to\infty$ has a solution exactly when $\beta>0$. Put $a=\beta^{1/3}>0$. Its unique real solution is
$$
v(\xi)=\frac{2}{\sqrt3a}\xi^{-a/2}\sin\left(\frac{\sqrt3a}2\log\xi\right).
$$
Indeed the <Euler differential operator> has characteristic roots $0,a,-a/2\pm i\sqrt3a/2$. Decay retains only the conjugate pair, and the two endpoint conditions determine their sine combination. For negative $\beta$ only one root has negative real part, so the two boundary conditions are incompatible. At zero every solution is a cubic polynomial in $\log\xi$, excluding nonzero decaying solutions. Complex exponents thus produce an oscillatory but decaying power-law envelope.
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