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Find f(x) that satisfies [f(x)]² + 4*f'(x)*f(x) + [f'(x)]² = 0.

1. f(x) = c * e^((2 - sqrt(3))*x)
2. f(x) = c * e^((2 + sqrt(3))*x)
3. f(x) = c * e^((sqrt(3) - 2)*x)
4. f(x) = c * e^((-2 + sqrt(3))*x)

Correct answer: f(x) = c * e^((sqrt(3) - 2)*x)

Solution

Assuming f = c*e^(kx) turns the equation into k² + 4k + 1 = 0, giving k = -2 +/- sqrt(3); the choice -2 + sqrt(3) = sqrt(3) - 2 matches an option.

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