Word Count : 496

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### Part 1: Riccati Equation associated with \( y'' + w^2 y = 0 \) To find the Riccati equation associated with \( y'' + w^2 y = 0 \), we begin by transforming the second-order linear differential equation into a first-order system. Assume a substitution for the first derivative of \( y \), namely: \[ \frac{dy}{dx} = p(x) \] Then, the second derivative of \( y \) is: \[ y'' = \frac{d^2y}{dx^2} = \frac{dp}{dx} \] Substituting into the equation \( y'' __________ _________ ___ _____ _______ ____ ___ _______.
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