Word Count : 386

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We are given the second-order differential equation: \[ \frac{d^2 y}{dx^2} - 2\frac{dy}{dx} + 10y = 0 \] with initial conditions: \[ y(0) = 4, \quad \frac{dy}{dx} (0) = 1. \] ### Step 1: Convert into a System of First-Order Equations Let: \[ y_1 = y, \quad y_2 = \frac{dy}{dx}. \] Then, \[ \frac{dy_1}{dx} = y_2. \] From the given differential equation: \[ \frac{d^2 y}{dx^2} = 2\frac{dy}{dx} - 10y. \] Using \( y_1 \) and \( y_2 \): \[ \frac{dy_2}{dx} = 2y_2 - 10y_1. \] Thus, we have the system: \[ \frac{dy_1}{dx} = y_2, \quad \frac{dy_2}{dx} = ____ _____ ______ ____ ________ _____.
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