= Solution
An <integrating factor> explains the second hint and the first-order inner calculation. For
$$
Y''+2zY'=2a_1z+a_2+a_3\operatorname{erf}z,
$$
multiplication by $e^{z^2}$ gives $(e^{z^2}Y')'=e^{z^2}(2a_1z+a_2+a_3\operatorname{erf}z)$. The term $a_1z$ already produces $2a_1z$, and integrating the remaining terms twice gives $a_2E_0+a_3E_1$, in terms of the <Gaussian drift primitives>. The <homogeneous solutions> are a constant and the <error function>. Hence
$$
\boxed{Y=a_1z+a_2E_0(z)+a_3E_1(z)+K_0+K_1\operatorname{erf}z.}
$$
For the piecewise forcing in the inner problem, $F$ replaces $a_1z$ and provides the continuously matched particular solution for $2zH(-z)$. The coefficients $2c$ and $2d$ then account for the $2Y_0$ forcing.
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