A differentiable function y = f(x) satisfies the relation f(x)*sin 2x - cos x + (1 + sin² x)*f'(x) = 0 together with the initial condition f(0) = 0. Which one of the following is correct?

Question

Accepted Answer

Range of f(x) is [-1/2, 1/2]. The equation is a first-order linear ODE in disguise. Since (1 + sin² x)' = sin 2x, the terms (1 + sin² x)*f' + f*sin 2x combine into the derivative of the product (1 + sin² x)*f(x). Integrating cos x gives sin x, and the initial condition kills the constant, yielding f(x) = sin x / (1 + sin² x). Substituting t = sin x in [-1, 1], the function t/(1 + t²) attains a maximum 1/2 at t = 1 and minimum -1/2 at t = -1, so the range is [-1/2, 1/2].

A differentiable function y = f(x) satisfies the relation f(x)sin 2x - cos x + (1 + sin² x)f'(x) = 0 together with the initial condition f(0) = 0. Which one of the following is correct?

Solution

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