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I'm currently reading through Gohberg and Goldberg's treatment of Sturm-Liouville systems in Basic Operator Theory.

Define a Sturm-Liouville system to be a differential equation of the form $$\frac{d}{dx}(p(x) \frac{dy}{dx}) + q(x) y = f(x)$$ with boundary conditions $$a_1 y(a) + a_2 y'(a) = 0$$ $$b_1 y(b) + b_2 y'(b) = 0$$ where $a_i, b_i$ are real numbers with $a_1^2 + a_2^2 \neq 0$, $b_1^2 + b_2^2 \neq 0$. Suppose the only solution where $f = 0$ is $y=0$. Then there exist real valued nonzero functions $y_1, y_2$ that are solutions, such that $y_1$ satisfies the first boundary condition and $y_2$ the second. Then they claim that a "straightforward computation" verifies that $(pW)' = 0$, where $W$ denotes the Wronskian. I do not see this computation. What I end up with is $$(pW)' = p'y_1 y_2' - p' y_2 y_1' + py_1' y_2' + py_1 y_2'' - py_2' y_1' - py_2 y_1''$$ and I don't see why it's zero.

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Wrewrite $(p\,W)'$ as $$(p\,W)'=y_1(p\,y_2')'-y_2(p\,y_1')' $$ and use the fact that $y_1$ and $y_2$ are solutions of the equation.

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