I have a list of word problems from my professor to review for an exam but he will not help us out at all. If anybody could answer any of these or even tell me how to start them I would greatly appreciate it!
Consider the following ODEs and the suggested change of variables. Try to solve the ODE without using the hint. If you are not successful, try it using
the hint.
(a) dy/dx= cos(x + y) Hint: let z = x + y .(b) (4x + 2y + 3)dy/dx + 2x + y + 1 = 0 Hint: let z = 2x + y
(c) 3(x + 2y)dy/dx + x + 2y-1 = 0 Hint: let z = x + 2y
(d) e^-y (dy/dx + 1) =x(e^x)
Fluctuating population: A model for a population that undergoes seasonal fluctuations (P(t) = population of the species) is given by the IVP
dP/dt= (k cos (omega)t)P ; t > 0 ; (1)
P(0) = P0
(a) Find the population P(t) at any time t , and check that it indeed solves the IVP.
(b) Choose P0 = 10, the period of fluctuations T = 2pi/(omega) = 10 units of time,
and k = 4 in units of time-1. Use Mathematica to plot the solution over 3 periods, namely for 0 < t< 3T .
Pharmacokinetics: Pharmaceutical companies need to understand how the concentration of certain drugs change throughout the body in order to deter-
mine dosage, toxicity, etc. This is usually done via a compartment model - the body is divided into compartments, and a balance law is formulated in each
compartment to account for the transport, consumption and elimination of the drug.
Consider a pill of antihistamine to be taken by a person. The pill goes to the GI tract where it dissolves and the antihistamine diuses through to the blood-
stream (BS), and it is then eliminated by chemical reactions or through the kidneys. In a simple model, we divide the entire body into two compartments:
the GI tract and the bloodstream. Let x(t) and y(t) be the concentrations of the drug in the GI tract and the bloodstream, respectively. Assume that the rate of clearance from each compartment is proportional to the drug concentration in that compartment. Call k1 and k2 the clearance coefficients for GI and BS, respectively.
(a) Formulate an IVP for the two concentrations.
(b) Let the initial concentrations be x(0) = A and y(0) = 0. Solve the IVP and check that the solution you found indeed solves the IVP. Note: You
will obtain two ODEs, one for x and one for y ; therefore, you will end up with an IVP for a system of ODEs.
Consider the following ODEs and the suggested change of variables. Try to solve the ODE without using the hint. If you are not successful, try it using
the hint.
(a) dy/dx= cos(x + y) Hint: let z = x + y .(b) (4x + 2y + 3)dy/dx + 2x + y + 1 = 0 Hint: let z = 2x + y
(c) 3(x + 2y)dy/dx + x + 2y-1 = 0 Hint: let z = x + 2y
(d) e^-y (dy/dx + 1) =x(e^x)
Fluctuating population: A model for a population that undergoes seasonal fluctuations (P(t) = population of the species) is given by the IVP
dP/dt= (k cos (omega)t)P ; t > 0 ; (1)
P(0) = P0
(a) Find the population P(t) at any time t , and check that it indeed solves the IVP.
(b) Choose P0 = 10, the period of fluctuations T = 2pi/(omega) = 10 units of time,
and k = 4 in units of time-1. Use Mathematica to plot the solution over 3 periods, namely for 0 < t< 3T .
Pharmacokinetics: Pharmaceutical companies need to understand how the concentration of certain drugs change throughout the body in order to deter-
mine dosage, toxicity, etc. This is usually done via a compartment model - the body is divided into compartments, and a balance law is formulated in each
compartment to account for the transport, consumption and elimination of the drug.
Consider a pill of antihistamine to be taken by a person. The pill goes to the GI tract where it dissolves and the antihistamine diuses through to the blood-
stream (BS), and it is then eliminated by chemical reactions or through the kidneys. In a simple model, we divide the entire body into two compartments:
the GI tract and the bloodstream. Let x(t) and y(t) be the concentrations of the drug in the GI tract and the bloodstream, respectively. Assume that the rate of clearance from each compartment is proportional to the drug concentration in that compartment. Call k1 and k2 the clearance coefficients for GI and BS, respectively.
(a) Formulate an IVP for the two concentrations.
(b) Let the initial concentrations be x(0) = A and y(0) = 0. Solve the IVP and check that the solution you found indeed solves the IVP. Note: You
will obtain two ODEs, one for x and one for y ; therefore, you will end up with an IVP for a system of ODEs.