4. a) Prove the following identity, which is also called Green's first identity: For every pair of functions f(x), g(x)

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4 A Prove The Following Identity Which Is Also Called Green S First Identity For Every Pair Of Functions F X G X 1 (36.84 KiB) Viewed 51 times

4. a) Prove the following identity, which is also called Green's first identity: For every pair of functions f(x), g(x) on (a, b), 12=b ["* ƒ"(x)g(x) dx = −¸ − ["* f'(a)}g'(x) dx + f'(x)\g(1) ** b) Use Green's first identity to prove the following result: If we have symmetric boundary condi- tions, and |x=b f(x)ƒ'(x) == <0 for all (real-valued) functions f(x) satisfying the BCs, then there is no negative eigenvalue. (Hint: Substitute f(x) = g(x) = X(x), a real eigenfunction.)