Necessary and Sufficient Condition for Quadratic Functional to be Positive Definite/Lemma 2

Theorem
Let $\map h x : \closedint a b \to \R$ be continuously differentiable $\forall x \in \closedint a b$.

Suppose the function $\map h x$ satisfies the equation:


 * $-\map {\dfrac \d {\d x} } {\paren {t P + \paren {1 - t} } h'} + t Q h = 0$

subject to the boundary conditions:


 * $\map h {a, t} = \map h {b, t} = 0$

Then:


 * $\displaystyle \int_a^b \paren {\paren {P h'^2 + Q h^2} t + \paren {1 - t} h'^2} \rd x = 0$