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Chapter 5: Q7E (page 271)
In Problems 7–9, solve the related phase plane differential equation (2), page 263, for the given system.
The solution is .
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The motion of a pair of identical pendulums coupled with a spring is modeled by the system
for small displacements (see Figure 5.36). Determine the two normal frequencies for the system.
In Problems 19–24, convert the given second-order equation into the first-order system by setting v=y’. Then find all the critical points in the yv-plane. Finally, sketch (by hand or software) the direction fields, and describe the stability of the critical points (i.e., compare with Figure 5.12).
In Problems 3 – 18, use the elimination method to find a general solution for the given linear system, where differentiation is with respect to t.
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’s 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.
The doubling modulo \({\bf{1}}\) map defined by the equation \(\left( {\bf{9}} \right)\)exhibits some fascinating behavior. Compute the sequence obtained when
Numbers of the form \({\bf{k/}}{{\bf{2}}^{\bf{j}}}\) are called dyadic numbers and are dense in \(\left( {{\bf{0,1}}} \right){\bf{.}}\)That is, there is a dyadic number arbitrarily close to any real number (rational or irrational).
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