Sturm-Liouville Problem/Unit Weight Function/Lemma

Theorem
Let $\map \alpha x: \R \to \R$ such that $\map \alpha x \in C^2 \closedint a b$.

Suppose:


 * $\displaystyle \forall \map h x \in C^2 \closedint a b: \map h a = \map h b = \map {h'} a = \map {h'} b = 0: \int_a^b \map \alpha x \, \map {h''} x \rd x = 0$

Then:


 * $\forall x \in \closedint a b: \map \alpha x = c_0 + c_1 x$

where $ c_0, c_1 \in \R $ are constants.

Proof
Let $ c_0, c_1 $ be defined by the conditions:


 * $\displaystyle \int_a^b \paren {\map \alpha x - c_0 - c_1 x} \rd x = 0$


 * $\displaystyle \int_a^b \rd x \int_a^x \paren {\map \alpha \xi - c_0 - c_1 \xi} \rd \xi = 0$

Suppose:


 * $\displaystyle \map h x = \int_a^x \xi \int_a^\xi \paren {\map \alpha t - c_0 - c_1 t} \rd t$

This form satisfies conditions on $h$ in the theorem.

Then:

On the other hand:

Therefore:


 * $\map \alpha x - c_0 - c_1 x = 0$

or:


 * $\map \alpha x = c_0 + c_1 x$