Differential Equations · Variable separable differential equations

Let

f : \mathbb{R} \to \mathbb{R}

be a twice differentiable function such that the quadratic equation

f(x)m^2 - 2f'(x)m + f''(x) = 0

m

, has two equal roots for every

x \in \mathbb{R}

. If

f(0) = 1

f'(0) = 2

and

(\alpha, \beta)

is the largest interval in which the function

f(\log_e x - x)

is increasing, then

\alpha + \beta

is equal to .

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