The function f(x) = x satisfies the following equation:

f''(x) + f'(x) + cos(x) = 0

The value of f''(0) is ________.

Question

The function f(x) = x satisfies the following equation:

f''(x) + f'(x) + cos(x) = 0

The value of f''(0) is ________.

Accepted Answer

-2 Since f(x) = x is given explicitly, we can directly compute its derivatives and substitute into the equation to find f''(0). But wait — f(x) = x gives f'(x) = 1 and f''(x) = 0, so 0 + 1 + cos(x) = 0 would require cos(x) = -1 everywhere, which is false. The question likely means that f is a general differentiable function whose value is zero at x = 0 (i.e. f(0) = 0) and satisfies the ODE, and we use the ODE directly at x = 0 to read off f''(0). Substituting x = 0: f''(0) + f'(0) + cos(0) = 0. We need f'(0). If f(0) = 0 and f(x) behaves like x near 0, then f'(0) = 1, giving f''(0) + 1 + 1 = 0, so f''(0) = -2.