Problems

1. Use Euler's Method to generate approximate values \(y_1\), \(y_2\), and \(y_3\) of the solution \(y(t)\) of the initial value problem

   \(\frac{dy}{dt}=t^2\)  with  \(y(0)=1\)

   and with the step size indicated in each part.

   a.   \(\Delta t = 0.1\)

   b.   \(\Delta t = 0.01\)

   c.   \(\Delta t = 0.001\)

2. Find the exact solution of the initial value problem

   \(\frac{dy}{dt}=t^2\)  with  \(y(0)=1\).

   [Hint: Determine all the functions that have derivative \(t^2\), and let \(y(t)\) be the one that satisfies the initial condition.]
3. Evaluate your solution function from Problem 2 at

   a.  \(t= 0.1,\,0.2,\,0.3\)

   b.  \(t= 0.01,\,0.02,\,0.03\)

   c.  \(t= 0.001,\,0.002,\,0.003\)

   Compare these exact values with the approximate values from Euler's Method in Problem 1. How good are the approximations?
4. Use Euler's Method to generate approximate values \(y_1\), \(y_2\), and \(y_3\) of the solution \(y(t)\) of the initial value problem

   \(\frac{dy}{dt}=t^2+1\)  with  \(y(0)=2\)

   and with the step size indicated in each part.

   a.   \(\Delta t = 0.1\)

   b.   \(\Delta t = 0.01\)

   c.   \(\Delta t = 0.001\)

5. Find the exact solution of the initial value problem

   \(\frac{dy}{dt}=t^2+1\)  with  \(y(0)=2\).

   [Hint: See Problem 2.]
6. Evaluate your solution function from Problem 5 at

   a.  \(t= 0.1,\,0.2,\,0.3\)

   b.  \(t= 0.01,\,0.02,\,0.03\)

   c.  \(t= 0.001,\,0.002,\,0.003\)

   Compare these exact values with the approximate values from Euler's Method in Problem 4. How good are the approximations?
7. A physician decides to give a patient an intravenous infusion of potassium (chemical symbol K) at a rate of 5 units per hour. The body of the patient simultaneously removes potassium from the bloodstream at 0.1 units per hour per unit of potassium present. There are 2.5 units of potassium in the bloodstream at time 0 — which is the reason for the infusion, since the normal range is 3.7 to 5.2 units.

   1. Explain why the amount \(K=K(t)\) of potassium present at time \(t\) can be modeled by the differential equation \(\frac{dK}{dt}=5-0.1K\).
   2. Find an explicit formula for the amount present at any time \(t\).
   3. At what time is the potassium level in the patient's bloodstream rising fastest? What is that fastest rate of increase?
   4. Suppose Euler's Method is used to generate approximate values of the glucose level, \(K_1\), \(K_2\), \(K_3\), and so on, at times \(t_1\), \(t_2\), \(t_3\), and so on, that are 15 minutes apart. What will the calculated approximate level of potassium be one hour after the infusion starts?
   5. Use your formula from part (b) to calculate the potassium level at the end of an hour. How does your approximation in part (d) compare with the potassium level calculated from the formula?

   Note 3 – Units for Potassium