Solve a first-order nonlinear ordinary differential equation (boundary value problem)

Add the derivative of the equation to the set of knowledge $$ [2y'(x^2-36)-2xy]y''=0 $$ This means that solutions can have segments where they are linear, $$ y=Cx+D, $$ where the constants may be connected via the original equation, and other segments where $$ \frac{y'}{y}=\frac{x}{x^2-36}\implies y^2=C(x^2-36), $$ where again $C$ might be bound by the original equation.

Insert these formulas and solve for the constants. Then try to find the correct combination of such pieces to satisfy the boundary condition. It might not be possible to get an everywhere continuous second derivative.

That the derivative factors so well is related to the fact that the reduction of the first equation via quadratic equation solution formula results in Clairaut equations.