Consider the ordinary differential equation

$$\frac{d^{2} \psi}{d z^{2}}-\left[\frac{15 k^{2}}{4(k|z|+1)^{2}}-3 k \delta(z)\right] \psi=0$$

where $k$ is a positive constant and $\delta$ denotes the Dirac delta function. Physically relevant solutions for $\psi$ are bounded over the entire range $z \in \mathbb{R}$.

(i) Find piecewise bounded solutions to this differential equations in the ranges $z>0$ and $z<0$, respectively. [Hint: The equation $\frac{d^{2} y}{d x^{2}}-\frac{c}{x^{2}} y=0$ for a constant $c$ may be solved using the Ansatz $y=x^{\alpha}$.]

(ii) Derive a matching condition at $z=0$ by integrating ( $\dagger$ ) over the interval $(-\epsilon, \epsilon)$ with $\epsilon \rightarrow 0$ and use this condition together with the requirement that $\psi$ be continuous at $z=0$ to determine the solution over the entire range $z \in \mathbb{R}$.