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Chapter 5: Q19E (page 271)

In Problems 19鈥�24, convert the given second-order equation into a first-order system by setting v=y鈥�. 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).

d2xdt2-y=0

Short Answer

Expert verified

The point is an unstable saddle point (0, 0).

Step by step solution

01

Find the critical point

Here the equation is d2xdt2-y=0.

Put v=y'鈥�鈥�and鈥�鈥�v'=y''.

Then the system is:

y'=vy''=yv'=y

For critical points equate the system equal to zero.

v=0y=0

So, the critical point is (0, 0).

The phase plane equation is:

dvdy=yv鈭�vdv=鈭�ydyv2-y2=c

02

Sketch

This is the required result.

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Most popular questions from this chapter

In Problems 19鈥�24, convert the given second-order equation into a first-order system by setting v=y鈥�. 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).

d2ydt2+y3=0.

Find the critical points and solve the related phase plane differential equation for the system dxdt=(x-1)(y-1),dydt=y(y-1).Describe (without using computer software) the asymptotic behavior of trajectories (as role="math" localid="1664302603010" t鈫�鈭�) that start at

  1. (3, 2)
  2. (2, 1/2)
  3. ( -2, 1/2)
  4. (3, -2)

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 Problems 3鈥�6, find the critical point set for the given system.

dxdt=x-y,dydt=x2+y2-1

In Problems 15鈥�18, find all critical points for the given system. Then use a software package to sketch the direction field in the phase plane and from this description the stability of the critical points (i.e., compare with Figure 5.12).

dxdt=2x+13y,dydt=-x-2y
See all solutions

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