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The Frobenius Method is a technique used to find power series solutions to linear differential equations around a singular point. Given the differential equation: \[ x^2 \frac{d^2 y}{dx^2} + (x + x^2) \frac{dy}{dx} + (x - 9)y = 0 \] we follow these steps to find the solution near \( x = 0 \): --- ### Step 1: Assume a Series Solution Since \( x = 0 \) is a regular singular point, we assume a Frobenius-type series solution: \[ y(x) = \sum_{n=0}^{\infty} a_n x^{n+r} \] where \( r \) is an unknown exponent to be determined. ### Step 2: Compute the Derivatives First derivative: \[ \frac{dy}{dx} _________ ______ _______ ________ ____ _________ __________ _________ _______ ________ _____ _________.
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