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We will solve the given initial value problem using the Laplace Transform. ### Step 1: Take the Laplace Transform of the Equation Given the differential equation: \[ y'' - 10y' + 9y = 5t \] Taking the Laplace Transform of both sides, using: \[ \mathcal{L}\{y''(t)\} = s^2 Y(s) - sy(0) - y'(0) \] \[ \mathcal{L}\{y'(t)\} = s Y(s) - y(0) \] \[ \mathcal{L}\{y(t)\} = Y(s) \] Substituting the given initial conditions: \( y(0) = -1 \) and \( ______ __________ _______ _____ ________ _________ _______ ______ ______ ______ ________ ___.
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