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

Theorem
If the function $ h \left ( { x } \right ) $ satisfies the equation


 * $ \displaystyle -\frac{ \mathrm d }{ \mathrm d x} \left [ { \left ( { t P + ( 1 - t) } \right ) h' } \right ]  + tQh = 0 $

and the boundary conditions


 * $ h \left ( { a, t } \right ) = h \left ( { b, t } \right ) = 0 $

then


 * $ \displaystyle \int_a^b \left [ { \left ( { P h'^2 + Q h^2 } \right )t + \left ( { 1 - t } \right ) h'^2 } \right ] \mathrm d x = 0$