During the diastolic phase of a heartbeat, the heart fills with blood, and no blood flows from the heart to the aorta. Assume that pressure P in the aorta is proportional to the volume V, with a constant of proportionality k. Assume further that the rate of change of the volume with respect to time t is proportional to the pressure, with the constant of proportionality -1/w.
a) Write down the differential equation describing how the pressure P changes with respect to time, and solve this differential equation with the condition that P = P0 at time t = 0.
b) During the systolic phase, blood from the heart enters the aorta. The rate that pressure changes with respect to time, described by the differential equation in part a, now has an added term due to the blood entering the aorta. Suppose this term can be described by A sin Bt for positive constants A and B. Solve this differential equation.