WKB form, large deviation expansion of the stationary PDF

Question

I am currently reading the book theory and applications of stochastic processes an analytical approach, on page 304, we have the equality: $$ \int_{\mathbb{R}} [p_{\varepsilon}(y-\varepsilon \xi) w(\xi \mid y - \varepsilon \xi) - p_{\varepsilon} (y) w(\xi\mid y)] d\xi = 0, \tag{1} $$ with $\varepsilon$ small. One then write the WKB form: $$ p_\varepsilon(y) \sim [K_0(y) + \varepsilon K_1(y) + \dots ] e^{-\psi(y)/\varepsilon} \tag2 $$ and notes that if substitute $(2)$ in $(1)$ and set all same-degree terms of $\varepsilon$ equal to zero one finds: $$ \int_{\mathbb{R}} \left[ e^{\xi \psi'(y)} - 1 \right] w(\xi \mid y,0 ) d\xi = 0 \tag3 $$ and $$ \int_\mathbb{R} \left( \frac{\partial}{\partial y} [w(\xi\mid y) K_0(y)] + \frac{\xi w(\xi\mid y)}{2} \psi''(y) K_0(y) \right) \xi e^{\xi \psi'(y)} d\xi = 0 \tag4 $$ but I don't really see how to derive $(3),(4)$ from $(1),(2)$. I think that if someone hints me how to solve $(3)$, $(4)$ should not be a problem.