Consider the following boundary value problem $$ \begin{cases} y''(x)=p(x)y'(x)+q(x)y(x)+r(x)\\ y(0)=\alpha,\quad y(1)=\beta. \end{cases} $$ for continuous $p(x)$, $q(x)$, $r(x)$ and positive $q(x)>0$.
According to Section 8.1 of "A friendly introduction to numerical analysis" by Brian Bradie solution exists based on convergence of finite-difference scheme.
Is there any other, more classical way to show solvability of this problem?
Edit: I figured it out, the equation can be rewritten as $(v(x)y'(x))'=q(x)v(x)y(x)+r(x)v(x)$ with $v'(x)=-p(x)v(x)$. In the result, the corresponding bilinear form would be coercive.