Consider a function $$ y = f(x) \text{ which satisfies the following equation:} $$ $$ \frac{d^2y}{dx^2} - \frac{dy}{dx} = 0. $$ As $ x \to -\infty, y = 1 $, and at $ x = 0, y = 2 $. The value of \[ \frac{dy}{dx} \text{ at } x = 0 \text{ is } _______ \text{ (answer in integer).}

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We are given the second-order linear differential equation: $$ \frac{d^2y}{dx^2} - \frac{dy}{dx} = 0. $$ To solve this, we first solve the homogeneous equation: $$ \frac{dy}{dx} = C_1 e^x. $$ Next, integrate to get: $$ y(x) = C_1 e^x + C_2. $$ Now, applying the boundary conditions: - As $ x \to -\infty $, $ y = 1 $, so $ C_2 = 1 $, - At $ x = 0 $, $ y = 2 $, so $ C_1 = 1 $. Thus, the solution is: $$ y(x) = e^x + 1. $$ Finally, compute $ \frac{dy}{dx} $ at $ x = 0 $: $$ \frac{dy}{dx} = e^x, $$ so at $ x = 0 $, $ \frac{dy}{dx} = 1 $. Thus, the value of $ \frac{dy}{dx} $ at $ x = 0 $ is $ 1 $.

Consider a function \[ y = f(x) \text{ which satisfies the following equation:} \] \[ \frac{d^2y}{dx^2} - \frac{dy}{dx} = 0. \] As \( x \to -\infty, y = 1 \), and at \( x = 0, y = 2 \). The value of \[ \frac{dy}{dx} \text{ at } x = 0 \text{ is } _______ \text{ (answer in integer).}

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