= Solution
Multiply the differential equation by $-p(x)$ and compare the coefficients of $y''$ and $y'$ with $-(py')'=-py''-p'y'$. The required <integrating factor> satisfies $p'=-bp$. Thus one may take
$$
\boxed{p(x)=\exp\left(-\int_{x_*}^x b(s)\,ds\right),\qquad q(x)=-p(x)c(x),\qquad w(x)=p(x)d(x).}
$$
Then $-(py')'+qy=\lambda wy$, the desired <Sturm-Liouville form>, with the same homogeneous <Dirichlet boundary conditions>. A common positive constant multiplying $p,q,w$ is immaterial. For the usual regular <Sturm-Liouville problem> one also assumes real sufficiently regular coefficients and $d>0$, ensuring $p,w>0$.
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