Kernel: SageMath 9.7

C23 Laplace Transforms Part 4

Exercise 5.4.5b. A salt tank contains 100 liters of pure water at time t=0t = 0, when salty water begins flowing into the tank at 2 liters per minute. The incoming liquid contains a concentration of 0.1 kg of salt per liter. The well-stirred liquid flows out of the tank at 2 liters per minute. At time t=20t = 20 minutes 5 kg of salt is dumped into the tank and dissolves instantaneously. Model the situation with a first-order ODE, with x(t)x(t) as the mass of salt in the tank at time tt. Solve this ODE using the Laplace transform. Plot the solution to make sure it’s sensible.

Exercise 5.1.1. A patient with no morphine in their body is given a 5 mg bolus of morphine at 9:00 AM, followed by infusion at a rate of 1 mg of morphine per hour. At 9:00 PM, the infusion rate is increased to 1.5 mg per hour. At 6:00 AM, a 5 mg bolus is administered and the infusion rate is decreased to 1 mg per hour. Formulate an appropriate differential equation, solve using Laplace transforms, and graph the amount of morphine in the patient over the first 24 hours.

Exercise 5.4.8. An unforced spring-mass-damper system obeys 2u(t)+4u(t)+52u(t)=02u''(t) + 4u'(t) + 52u(t) = 0 where u(t)u(t) is the position of the mass, with initial conditions u(0)=1u(0) = 1 and u(0)=1u' (0) = −1.

(a) Solve the ODE to find the position u(t) of the mass.

(b) The system is underdamped, so the mass repeatedly passes through equilibrium. Find the second positive time t=t2t = t_2 at which the mass passes through equilibrium.

(c) Suppose that at time t=t2t = t_2 a hammer blow of impulse AA is to be applied to the mass to bring it to a dead stop. What should AA equal? (AA can be negative.) Hint: it should counteract the momentum of the mass.

(d) Solve the ODE 2u(t)+4u(t)+52u(t)=Aδ(tt2)2u''(t) + 4u'(t) + 52u(t) = Aδ(t −t_2) with initial conditions u(0)=1u(0) = 1 and u(0)=1u'(0) = −1,and AA and t2t_2 as from parts (b) and (c). Plot the solution on the range 0t20 ≤ t ≤ 2 .