2 CBEMS 230 Fall 2016 Due Tuesday, October 11 1. We are interested in predicting the concentration of a hormone in a patient's blood as a function of time. Suppose that the hormone has a periodic input rate with a period of 24 hours into the blood stream from the thyroid gland. Also, the hormone is being continuously removed at a rate that is proportional to its concentration. Therefore, we can write: \u0012 Input = A + Bcos 2t 24 \u0013 , Output = ky(t) where A, B, and k are constants, and y(t) denotes the hormone concentration in the blood. (a) Write down a differential equation governing the hormone concentration in the patient's blood. (b) Obtain the general solution of the ODE from part (a). (Hint: you can use the method of integrating factor. You can also consult integral tables if needed). 2. Find the solutions to the following ODEs. (a) y 00 + y = x sin(x) (b) y 00 + y = 0.001x2 , y(0) = 0, y 0 (0) = 1.5 (c) y 00 + 3y 0 + 2.25y = 10 exp(1.5x), y(0) = 1, y 0 (0) = 0 3. Consider the harmonic oscillator shown in the Figure, in which a body of mass m is attached to a wall on one end via a spring with spring constant k, and on the other end to a dashpot (mechanical damping device) with damping constant c. The spring and dashpot are assumed to have negligible masses compared to m. At equilibrium, 1 Figure 1: Harmonic Oscillator the body rests at y = 0, and downward displacement is considered to be positive. If we pull the body down a certain distance y0 and release it at t = 0, it will move in a way that depends on the spring stiffness and the damping effect of the dashpot. To find the position of the body as a function of time, y(t), a force balance must be written. The forces acting on this body are the spring force, which is proportional to the stretch distance and in the opposite direction, Fs = ky, and the damping force, which is proportional to the velocity, and again in the opposite direction: Fd = cy 0 . We can neglect gravity. The net force then gives rise to acceleration of the body, F = my 00 . If there are no other forces acting on the mass, then the differential equation governing the motion of this body is my 00 = ky cy 0 . (a) Consider m = 10kg, k = 90kg/s2 , and c = 100kg/s. If the body is pulled down to y0 = 16cm and released with zero initial velocity, solve the ODE. (Hint: because we have specified initial conditions, there should be no arbitrary (unspecified) constants left in your final solution). (b) Use software to plot the position of the body for 0 t 10s. (c) Solve and then plot (on the same graph as part (b)) the solution if c = 10kg/s and everything else is the same as part (b). Comment on the difference between the curves. 4. Consider a flask (total volume 500 ml) initially containing 200 ml of a solution that has 100 g salt in it. The flask has an inlet and outlet; at t = 0, salt solution begins to flow continuously in (inlet stream concentration Cin = 1g/ml, flow rate V in = 3ml/min), and well-mixed solution begins to flow out (output flow rate V out = 2ml/min). 2 (a) Draw a schematic cartoon of this problem, to help you solve it. (b) Find an equation that describes the amount of salt (in g) in the flask at any time 0 < t < tover , where tover is the instant of overflow. (c) Plot the concentration of salt in the tank C(t) for 0 < t < tover . 3