Consider the solution to the Painlevé II equation on $\mathbb{R}$
$$q''=2q^3+rq$$
with the boundary condition $q(r)\sim_{r\to +\infty} \mathrm{Ai}(r)$ and consider the function $f$ such that for $r\in \mathbb{R}$,
$$f(r)=\cosh\left(\frac{1}{2}\int_r^{+\infty} \mathrm{d} s \, q(s)\right)$$
We have that $f$ verifies the third order differential equation
$$f'''+P(r) f' +Q(r) f =0 \qquad \qquad (\mathcal{E})$$
with
$$\begin{cases} P(r)=-\frac{9}{4}q(r)^2-r\\ Q(r)=\frac{1}{6}P'(r)+\frac{1}{6} \end{cases}$$
My question is the following, if one is given the differential equation $(\mathcal{E})$ with the proper boundary condition, is there a standard technique / change of variable to find back the expression of the function $f$ ?