I have been reading a book by Jeanblanc, Yor, Chesney (2009) "Mathematical Methods for Financial Markets" and ran into the section on resolvent kernels which is very terse (Section $5.3.6$, page $277$). I do not understand the connection between resolvent kernel of a diffusion (denoted below by $R_{\lambda}(x,y)$) and the following ODE:
$\frac{1}{2} \sigma^2(x)\frac{\partial^2R_{\lambda}}{\partial x^2} + b(x) \frac{\partial R_{\lambda}}{\partial x} - \lambda R_{\lambda}=0$
where $d X_t = b(X_t)dt+\sigma(X_t)dW_t$ is a diffusion process and $W_t$ denotes a standard brownian motion
Here is the excerpt from the book that I am trying to understand (below $X_t$ is a time homogeneous diffusion proces satisfying $d X_t = b(X_t)dt+\sigma(X_t)dW_t$ with $X_0=x$).
Resolvent Kernel
The resolvent of a Markov process X is a family of operators $f \rightarrow R_{\lambda}f$ $$ R_{\lambda}f(x)=\mathbb{E}_x\left(\int_0^{\infty} e^{\lambda t} f(X_t) dt \right)$$ The resolvent kernel of a diffusion is the density (with respect to Lebesgue measure) of the resolvent operator, i.e., the Laplace transform in t of the transition density $p_t(x,y)$ : $$ R_{\lambda}(x,y) = \int_0^{\infty} e^{-\lambda t} p_t{(x,y)}dt$$ It satisfies $$ \frac{1}{2} \sigma^2(x)\frac{\partial^2R_{\lambda}}{\partial x^2} + b(x) \frac{\partial R_{\lambda}}{\partial x} - \lambda R_{\lambda}=0 $$ and $R_{\lambda}(x,x)=1$.
Question: How to obtain that ODE from the definition of $R_{\lambda}(x,y)$?
This is probably trivial (so I apologize in advance if that is indeed the case), but since I am self-studying this there is nobody I can consult.
A related issue: The authors then use the solutions to this equation to obtain expressions for the Laplace transforms of the first hitting time in terms of solutions to the above ODE. $$ \mathbb{E_x}(e^{-\lambda T_{y}}) = \Phi_{\lambda \uparrow}(x)/\Phi_{\lambda \uparrow}(y) \text{ if x<y}$$ where $\Phi_{\lambda \uparrow}(x)$ is an increasing function that satisfies the above ODE. A result which I also have hard time understanding...