Consider the inhomogeneous ordinary differential equation

(4.128)

$$\frac{d^{2}u_{k}(z)}{dz^{2}}-(j_{k}^{2}z^{2}+\lambda_{k})u_{k}(z)=\xi_{k}(z)\cdot z,\ z\geq 10^{6},$$

where $\underline{\delta}>0$ is the constant defined in (4.106). Assume that the function $\xi_{k}(z)$ satisfies the following property: there are constants

(4.129)

$$\eta_{0}\in(-\underline{\delta}/2,\underline{\delta}/2)\setminus\{0\}$$

and $Q_{k}>0$ such that

(4.130)

$$|\xi_{k}(z)|\leq Q_{k}\cdot e^{\eta_{0}z}.$$

Let $u_{k}(z)$ be the particular solution defined by

(4.131)

$$u_{k}(z)\equiv\frac{\mathcal{G}_{k}(z)+\mathcal{D}_{k}(z)}{\mathcal{W}_{k}(z)},$$

where

(4.132)

$$\mathcal{D}_{k}(z)\equiv\mathcal{F}_{k}(z)\int_{z}^{\infty}\mathcal{U}_{k}(r)\cdot\Big(\xi_{k}(r)\cdot r\Big)dr,$$

(4.133)

$$\mathcal{G}_{k}(z)\equiv\mathcal{U}_{k}(z)\int_{1}^{z}\mathcal{F}_{k}(r)\cdot\Big(\xi_{k}(r)\cdot r\Big)dr$$

and $\mathcal{W}_{k}$ is the Wronskian

(4.134)

$$\mathcal{W}_{k}(z)\equiv\mathcal{W}\Big(\mathcal{F}_{k}(z),\mathcal{U}_{k}(z)\Big).$$

Then there are constants $C_{0}>0$ and $\eta_{0}<\eta<\eta_{0}+\underline{\delta}/10$ which are independent of $k$ such that the particular solution $u_{k}$ satisfies the uniform estimate

(4.135)

$$|u_{k}(z)|\leq C_{0}\cdot Q_{k}\cdot e^{\eta z}.$$