Question

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).}

Hint:

For solving differential equations, first solve the general solution, then apply the boundary conditions to find the constants.

Answer 1

Correct Answer: 1

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 \( 1 \).