Sturm-Liouville Problem/Unit Weight Function/Lemma

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Theorem

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

Suppose:

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

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:

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

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

Suppose:

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

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

Then:

\(\ds \int_a^b \paren {\map \alpha x - c_0 - c_1 x} \map {h} x \rd x\)

\(=\)

\(\ds \int_a^b \map \alpha x \map {h} x \rd x - c_0 \paren {\map {h'} b - \map {h'} a} - c_1 \int_a^b x \map {h} x \rd x\)

\(\ds \)

\(=\)

\(\ds -c_1 \paren {b \map {h'} b - a \map {h'} a} - c_1 \paren {\map h b - \map h a}\)

\(\ds \)

\(=\)

\(\ds 0\)

On the other hand:

\(\ds \int_a^b \paren {\map \alpha x - c_0 - c_1 x} \map {h} x \rd x\)

\(=\)

\(\ds \int_a^b \paren {\map \alpha x - c_0 - c_1 x}^2 \rd x\)

\(\ds \)

\(=\)

\(\ds 0\)

Therefore:

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

or:

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

$\blacksquare$