Two differentiable functions f and g on R⁺ satisfy x f'(x) + g(x) = 0 and x g'(x) + f(x) = 0 for every x 0, with f(1) + g(1) = 4. Find the value of f''(2) * g''(2).

Question

Two differentiable functions f and g on R⁺ satisfy x f'(x) + g(x) = 0 and x g'(x) + f(x) = 0 for every x > 0, with f(1) + g(1) = 4. Find the value of f''(2) * g''(2).

Accepted Answer

1/4. Solving the decoupled ODEs gives f+g = 4/x and f-g = c x; the linear term has zero second derivative, so f''(2) = g''(2) = 1/2 and their product is 1/4.

Two differentiable functions f and g on R⁺ satisfy x f'(x) + g(x) = 0 and x g'(x) + f(x) = 0 for every x > 0, with f(1) + g(1) = 4. Find the value of f''(2) * g''(2).

Solution

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