Weierstrass-Erdmann Corner Conditions

Consider the functional \begin{align} J[y] = \int_{a}^{b}(y'^2+2xy-y^2)\,dx \end{align} subject to the boundary conditions \begin{align} y(a)=y_a, \ y(b) = y_b. \end{align} What kind of corner solutions exist for this problem?

Letting $$f(x,y,y') = y'^2+2xy-y^2$$, I solve the DE to get \begin{align} y = c_1cos(x)+c_2sin(x)+x, \end{align} but are there even any corner solutions for this?

Any tips would be appreciated!