= Radial modified Helmholtz equation
For a <radial function> in three dimensions, $(\Delta-1)y=g(x)$ becomes $y''+2y'/x-y=g(x)$. Setting $u=xy$ gives $u''-u=xg(x)$, reducing it to a constant-coefficient <linear ordinary differential equation>. If $g=e^{-3x}/x^2$, a decaying particular solution obtained by <variation of parameters> is
$$
y(x)=\alpha\frac{e^{-x}}x+\frac{e^{-x}E_1(2x)-e^xE_1(4x)}{2x}.
$$
The <small-argument expansion of the exponential integral> gives
$$
y(x)=\frac{\alpha+\tfrac12\log2}{x}+\log x-\alpha+\tfrac32\log2+\gamma-1+O(x\log x).
$$
The undetermined decaying homogeneous coefficient $\alpha$ is fixed by a boundary condition or by a <matched asymptotic expansion>.
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