1. Solve the differential equations first for the general solution as a function of y, and then for the initial condition. a) y' = x csc y b) y' = xy 2 In y y(T) = 1/3 y(0) = 1 3. Use Euler's method with step size 0.1 to estimate y(0.5), where y(x) is the solution of the initial-value problem y' = 2y-x, y(0) = 1 5. The rate of change of atmospheric pressure P with respect to altitude h is proportional to P, provided that the temperature is constant. At 15C, the pressure is 101.3 kPa at sea level (h = 0) and 87.14 kPa at h = 1000 m. a) Write an expression that gives the pressure P as a function of altitude, h. b) What is the pressure at an altitude of 2000 m? c) At what altitude will the pressure drop to 50 kPa? u-substitution 2. Solve the differential equations first for the general solution as a function of y, and then for the initial condition. a) x(ax) + 2xy = x, x>0 sin x b) == dx X X y(1) = 0 .4 - Y(TT) = 4 4. A tank contains 100 L of water. A solution with a salt concentration of 0.4 kg/L is added at a rate of 5 L/min. The solution is kept mixed and is drained from the tank at the same rate. a) If y(t) is the amount of salt (in kilograms) after t minutes, write and solve a differential equation describing this situation. b) Find the concentration after 20 minutes. c) What happens to the concentration of salt as t? 6. Suppose a series circuit has R = 15 Q, L = 3 H and a constant voltage E(t) = 50 V. Its current can be modeled by the differential equation L() + RI = E(t). (Assume I(0)=0) dt a) Find I(t) b) Find the current after 0.1 seconds. c) What is the limiting value of this current? 7. A population of rabbits in a meadow is observed to be 200 rabbits at time t = 0, where t is measured in months. After a month, the rabbit population is observed to have increased by 4. Suppose the carrying capacity for this population is 750. a) Assuming that the size of the rabbit population satisfies the logistic equation, find an expression for the size of the population after t months. b) How many rabbits are in the population after one year? c) How long will it take for the population to increase to 500?