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Chapter 5: Q9E (page 271)
In Problems 7鈥�9, solve the related phase plane differential equation (2), page 263, for the given system.
The solution is.
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Figure 5.16 displays some trajectories for the system What types of critical points (compare Figure 5.12 on page 267) occur at (0, 0) and (1, 0)?
Radioisotopes and Cancer Detection. A radioisotope commonly used in the detection of breast cancer is technetium-99m. This radionuclide is attached to a chemical that upon injection into a patient accumulates at cancer sites. The isotope鈥檚 radiation is then detected and the site is located, using gamma cameras or other tomographic devices.
Technetium-99m decays radioactively in accordance with the equation\(\frac{{{\bf{dy}}}}{{{\bf{dt}}}}{\bf{ = - ky}}\) with k= 0.1155>h. The short half-life of technetium-99m has the advantage that its radioactivity does not endanger the patient. A disadvantage is that the isotope must be manufactured in a cyclotron. Since hospitals are not equipped with cyclotrons, doses of technetium-99m have to be ordered in advance from medical suppliers.
Suppose a dosage of 5 millicuries (mCi) of technetium- 99m is to be administered to a patient. Estimate the delivery time from production at the manufacturer to arrival at the hospital treatment room to be 24 hours and calculate the amount of the radionuclide that the hospital must order, to be able to administer the proper
dosage.
In Section 3.6, we discussed the improved Euler鈥檚 method for approximating the solution to a first-order equation. Extend this method to normal systems and give the recursive formulas for solving the initial value problem.
In Problems 1鈥�7, convert the given initial value problem into an initial value problem for a system in normal form.
[hint]
Secretion of Hormones.The secretion of hormones into the blood is often a periodic activity. If a hormone is secreted on a 24-h cycle, then the rate of change of the level of the hormone in the blood may be represented by
the initial value problem\(\frac{{{\bf{dx}}}}{{{\bf{dt}}}}{\bf{ = \alpha - \beta cos}}\frac{{{\bf{\pi t}}}}{{{\bf{12}}}}{\bf{ - kx,x(0) = }}{{\bf{x}}_{\bf{o}}}\)where x(t) is the amount of the hormone in the blood at the time t, \({\bf{\alpha }}\) is the average secretion rate, \({\bf{\beta }}\)is the amount of daily variation in the secretion, and kis a positive constant reflecting the rate at which the body removes the hormone from the blood. If \({\bf{\alpha }}\)=\({\bf{\beta }}\) = 1, k= 2, and \({{\bf{x}}_{\bf{o}}}\) = 10, solve for x(t).
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