Solving $y'' + (ax+b)y = 0$

The solution is in term of the Airy functions, $\text{Ai}(x)$, $\text{Bi}(x)$ (which are well defined, have asymptotic formulas and series representation, etc. see for example Abramowitz & Stegun or Szegö).

These functions are the linearly indepedent solutions of $$\frac{d^2y(x)}{dx^2}-xy(x)=0 $$ With the change of variable $$\tilde{x} =\mu x +\nu$$ you will be able to transform the initial form to a Airy form. The solution is then directly

$$y(x)=\alpha \text{Ai}(\mu x + \nu)+\beta \text{Bi}(\mu x + \nu) $$

where $\alpha,\beta$ are the integration constants.

I) OP considers the TISE

$$ \hbar^2\psi^{\prime\prime}(x) +\alpha (x\!-\!x_0)\psi(x)~=~0, \qquad \alpha~\neq~0,\tag{1}$$

with an affine potential, i.e. the Airy differential equation (DE). We need to find 2 independent solutions. Here $\alpha\in \mathbb{R}\backslash\{0\}$ is a non-zero real constant, and $x_0$ is a turning point, i.e. the boundary point between the classically allowed and forbidden $x$-regions.

II) OP's idea is good. If we Fourier transform

$$\tilde{\psi}(p) ~=~ \int_{\mathbb{R}} \! \frac{\mathrm{d}x}{\sqrt{2\pi \hbar}}~\exp\left( -\frac{ipx}{\hbar} \right) \psi(x) \tag{2}$$

the source-free/homogeneous second-order ODE (1), we get a source-free/homogeneous first-order ODE

$$ i\hbar\tilde{\psi}^{\prime}(p) - \left(x_0+\frac{p^2}{\alpha}\right)\tilde{\psi}(p)~=~ 0,\tag{3} $$

with solution

$$ \tilde{\psi}(p)~=~\exp\left( -\frac{i}{\hbar} \tilde{S}(p)\right),\qquad \tilde{S}(p)~:=~p\left(x_0+\frac{p^2}{3\alpha}\right),\tag{4} $$

up to an overall normalization factor.

III) Eqs. (2) and (4) suggest that we should consider the contour integral

$$ \psi(x)~=~\int_{\gamma} \! \frac{\mathrm{d}p}{\sqrt{2\pi \hbar}}~\exp\left(\frac{i}{\hbar} \left(px-\tilde{S}(p)\right)\right), \tag{5} $$

where $\gamma$ is an open contour between exponentially suppressed sectors $|p| \to \infty$ in the complex $p$-plane, such that the integral (5) is well-defined and convergent.

$\uparrow$ Fig. 1. The complex $p$-plane with 3 possible integration contours ${\cal C}_1$, ${\cal C}_2$, ${\cal C}_3$ in the case $\alpha<0$. The shaded regions denote exponentially decaying sectors. (Figure taken from Ref. [W].)

IV) The contour integral (5) indeed satisfies the Airy DE (1):

$$ \hbar^2\psi^{\prime\prime}(x) +\alpha (x\!-\!x_0)\psi(x) ~\stackrel{(5)}{=}~\int_{\gamma} \! \frac{\mathrm{d}p}{\sqrt{2\pi \hbar}}~\left(-p^2 +\alpha (x\!-\!x_0)\right) \exp\left(\frac{i}{\hbar} \left(px-\tilde{S}(p)\right)\right)$$ $$~=~-i\hbar\alpha \int_{\gamma} \!\frac{\mathrm{d}p}{\sqrt{2\pi \hbar}}~\frac{d}{dp}\exp\left(\frac{i}{\hbar} \left(px-\tilde{S}(p)\right)\right)~=~0.\tag{6} $$

Since the integrand (5) is an entire function in $p$, the $p$-integral (5) only depend on the monodromy of the contour. Since there are 3 exponentially decaying sectors, there are 2 independent contours, leading to 2 independent solutions (5) to the Airy DE (1). Particular choices of the contour $\gamma$ lead to integral representations for the Airy functions ${\rm Ai}$ and ${\rm Bi}$ up to affine scaling.

References:

• [W] E. Witten, Analytic Continuation Of Chern-Simons Theory, arXiv:1001.2933; p. 23-29, 48-49. A related 2015 KITP lecture by Witten, A New Look At The Path Integral Of Quantum Mechanics, can be found on YouTube.