The question I am having trouble with is # 20, however it requires the previous

Question

The question I am having trouble with is # 20, however it requires the previous problem. I included both and the answer for the previous question.

19. A patient initially has 100 mg and 50 mg of morphine, respectively, in her blood and cerebrospinal fluid compartments. The per capita rates at which the drug is transferred across the blood-brain barrier from blood to cerebrospinal fluid and vice versa are a = 1/2 and b = 1 /4. The rates of metabolism of the drug within these pools are c = 1/5 and d 1 /10, respectively. Find the equilibrium solution and the particular solution to this problem. Use the particular solution to calculate the concentrations of morphine in the blood and the cerebrospinal fluid after two hours ANSWER WORKED SOLUTION Let x (t) be the amount of drug in the blood, and y (r) be the amount of drug in the cerebrospinal fluid. The equations describing the process are x = y' = x/2-y /4-y/10. This corresponds to the system u-Auwhere A = and corresponding eigenvectors are approximately 1-0.919. -=-0.131and x/2+y/4 x15 and 100 50 The equilibrium solution is given by u- eigenvalues - 1.139 and 0.439 , respectively. The initial conditions give the solution 56.34e-0919, - 49.46e -0919 +43.66e -0.131 Thus after two hours, u 2 42.59 4946999.4 Taus a io boarn -6273 (mg) -0.9 19t + 99.46e-0.131, 1 . Thus a 68.75 (mg,) 20. Suppose the patient in Problem 19 initially has no morphine in her system, but morphine is infused into her blood at a rate of 10 mg per hour. Write a model of this process, calculate the equilibrium solution, and calculate the particular solution for the initial condition of no morphine in the body

Explanation / Answer

Let x(t) be the amount of drug in the blood.

Also given that patient has initially no morphine in the blood. So x(0) = 0

Morphine is infused into her blood at a rate of 10mg/hr

So x'(t) = 10mg/hr.

So dx/dt = 10 ==> dx = 10dt

solving this by taking integration on both sides, we get x(t) = 10t + c

but for t = 0 , we know x(0) = 0

So x(0) = 10(0) + c ==> c = 0

Therefore, x(t) = 10t

So the particular solution is x(t) = 10t when no morphine in the system.

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