Question

If \( y = Ae^x + B \) where \( A, B \) are constants, then show that \( \frac{d^2y}{dx^2} - \frac{dy}{dx} = 0 \).

Hint:

If a function satisfies a differential equation after substitution, it is a valid solution.

Answer 1

Step 1: Differentiate \( y = Ae^x + B \). \[ \frac{dy}{dx} = A e^x \] Step 2: Differentiate again. \[ \frac{d^2y}{dx^2} = A e^x \] Step 3: Substitute in given equation. \[ \frac{d^2y}{dx^2} - \frac{dy}{dx} = A e^x - A e^x = 0 \] Thus, the function satisfies the equation.