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We are asked to solve the second-order differential equation numerically, manually, for: $$ \frac{d^2y}{dx^2} - 2 \frac{dy}{dx} + y = 0, \quad y(0) = 1, \quad y'(0) = 3 $$ Let's solve this step by step. --- ### Step 1: Convert to a system of first-order equations Let $$ y_1 = y, \quad y_2 = \frac{dy}{dx} = y' $$ Then the system becomes: $$ \frac{dy_1}{dx} = y_2 $$ $$ \frac{dy_2}{dx} = 2 y_2 - y_1 $$ With initial conditions: $$ y_1(0) = 1, \quad y_2(0) = 3 $$ --- ### Step __________ ___ _________ _________ _______ ______.
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