= Mészáros equation solution basis
{c}
{title2=$D_+(y),D_-(y)$}
A <basis> of the <solution space of a homogeneous linear differential equation> for the <Mészáros equation> on $y>0$ is
$$
D_+=1+\frac32y,\qquad
D_-=\left(1+\frac32y\right)\log\frac{\sqrt{1+y}+1}{\sqrt{1+y}-1}-3\sqrt{1+y}.
$$
Substitution verifies both solutions; their <Wronskian> is $-1/[y\sqrt{1+y}]$, so they are independent. At small $y$, $D_+=1+O(y)$ and $D_-=\log(4/y)-3+O(y\log y)$. At large $y$, $D_+\sim3y/2$ and $D_-\sim4/(15y^{3/2})$. Matching the actual early-time radiation forcing determines which constant and logarithmic combination is excited.
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