= Solution
The sign of the <Dirac delta distribution> in the equation is positive after the outer minus sign is expanded. Integrating through zero, the nonsingular potential contributes a term tending to zero, whereas the delta term contributes $3k\psi(0)$. The <jump condition for an impulse> is therefore
$$
\boxed{\psi'(0^+)-\psi'(0^-)=-3k\psi(0).}
$$
<Continuity> at zero gives $C_+=C_-=C$. The two one-sided derivatives are $-3kC/2$ and $+3kC/2$, whose difference is exactly $-3kC$. Hence the matching condition introduces no further restriction on $C$, and the globally bounded solution is
$$
\boxed{\psi(z)=C(1+k|z|)^{-3/2},\qquad C\in\mathbb R\ \text{or}\ \mathbb C.}
$$
This continuous solution has precisely the derivative jump required for its second <distributional derivative> to cancel the delta term. Its amplitude remains arbitrary because no normalization has been imposed.
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