Name: 1. In a chemical reaction, the rate at which a reactant diminishes (i.e., combines with another reactant) is proportional to the amount remaining. Let A(t) denote the amount of the reactant at time t. (a) (5 points) Use a di'erential equation to model the amount of the reactant in time. (b) (10 points) Solve your DE model. (c) (5 points) Suppose an initial amount of 10 grams of reactant decays to 6 grams in 2 minutes. Determine any parameter values using this data. (d) (5 points) Use your model (solution) to predict how long it would take for the reactant to decay to 1 gram. 5. (15 points) Sketch the phase line and sample solution curves for at = y(1 - y?) and determine the stability of any equilibrium values. 6. (15 points) Solve the IVP y" + 4y' + 3y = 0, y'(0) = 1, y(0) = 0. Determine the long-term behavior of the solution, i.e., find limt->co I(t). Also include a graph of the solution.7. (10 points) Find the general solution of y" 2y' + 4y = 0. 8. (5 points) 13 e" even real? Prove it