Ord Diff Eqs HW#01 Solutions Page 1 of 14 Section 1.1: #1, 5, 7, 9, 11, 13, 15, 16, 17, 18, 19, 20, 21, 22, 25, 27, and 31.

1) Draw a direction field for , determine the behavior as , and state how this behavior may depend on the initial value of y at t = 0.

Look for an equilibrium solution by taking:

From the direction field, it appears that the slopes are negative for y > 1.5, so any initial value y(0) > 1.5 converges to y = 1.5. Also, slopes appear to be positive for y < 1.5, so any initial value y(0) < 1.5 also converges to y = 1.5. So, as , the behavior of all solutions is to converge to y = 1.5.

Ord Diff Eqs HW#01 Solutions Page 2 of 14 Section 1.1: #1, 5, 7, 9, 11, 13, 15, 16, 17, 18, 19, 20, 21, 22, 25, 27, and 31.

5) Draw a direction field for , determine the behavior as , and state how this behavior may depend on the initial value of y at t = 0.

Look for an equilibrium solution by taking:

From the direction field, it appears that the slopes are positive for y > -0.5, so any initial value y(0) > -0.5 diverges away from y = 1.5. Also, slopes appear to be negative for y < -0.5, so any initial value y(0) < -0.5 also diverges from y = 1.5. So, as , the behavior of all solutions is to diverge away from the equilibrium solution y = -0.5.

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7) Write a differential equation of the form whose solution has the behavior that all solutions

approach y = 3 as . Choose a < 0 for convergence, and for the correct value of the equilibrium solution, we need:

one choice is

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9) Write a differential equation of the form whose solution has the behavior that all other solutions

diverge from y = 2 as . Choose a > 0 for divergence, and for the correct value of the unstable equilibrium solution, we need:

one choice is

Ord Diff Eqs HW#01 Solutions Page 5 of 14 Section 1.1: #1, 5, 7, 9, 11, 13, 15, 16, 17, 18, 19, 20, 21, 22, 25, 27, and 31.

11) Draw a direction field for , determine the behavior as , and state how this behavior may depend on the initial value of y at t = 0.

Look for equilibrium solutions by taking:

From the direction field, it appears that for initial conditions y(0) > 0, all solutions converge to the equilibrium solution of y = 4. For initial conditions y(0) < 0 all solutions appear to diverge to -.

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13) Draw a direction field for , determine the behavior as , and state how this behavior may depend on the initial value of y at t = 0.

Look for equilibrium solutions by taking:

From the direction field, it appears that for initial conditions y(0) < 0, all solutions converge to the equilibrium solution of y = 0. For initial conditions y(0) > 0 all solutions appear to diverge to +.

15) Stable equilibrium at y = 2 this is equation (j)

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16) Unstable equilibrium at y = 2 this is equation (c)

17) Stable equilibrium at y = -2 this is equation (g)

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18) Unstable equilibrium at y = -2 this is equation (b)

19) Unstable equilibrium at y = 0 and stable equilibrium at y = 3 this is equation (h)

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20) Unstable equilibrium at y = 3 and stable equilibrium at y = 0 this is equation (e)

21) A pond initially contains 1,000,000 gallons of an undesirable chemical; water with 0.01 g per gallon of this chemical flows in at a rate of 300 gal/hr. Water flows out at the same rate, so the volume of water remains constant (and the chemical mixes instantaneously throughout the pond)(a) Differential equation for amount of chemical in pond at any time:

Let M(t) be the amount (Mass) of chemical (measured in grams) in the pond as a function of time t (measured in hours)

The amount of chemical that flows into the pond is

The amount of the chemical that flows out will be the concentration of the chemical times the volume that

flows out:

The rate of change of the chemical will be what flows in minus what flows out:

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(b) The amount of chemical in the pond after a very long time will be:

From the direction field, it appears that this will be the amount after a very long time no matter how much is in the pond to begin with:

22) Write a differential equation for the volume of a raindrop if it evaporates at a rate proportional to its surface area. Let V(t) be the volume of the drop as a function of time, and let A(t) be the area as a function of time.

;

The differential equation is for k > 0

where the constant a combines the constant k and the numeric constants.

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25) For rapidly falling bodies, it is more correct to model the air drag as being proportional to the square of the velocity.(a) Write a differential equation:

Instead of . We would now have:

(b) Limiting velocity:

(c) If m = 10 kg, find drag coefficient so that limiting velocity is 49 m/sec

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(d) Draw a direction field and compare with linear drag:

Note: the quadratic drag force results in faster convergence to the equilibrium solution.

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27) Draw a direction field for , determine the behavior as , and state how this behavior may depend on the initial value of y at t = 0.

All solutions appear to approach the equilibrium solution of y = 0.

Ord Diff Eqs HW#01 Solutions Page 14 of 14 Section 1.1: #1, 5, 7, 9, 11, 13, 15, 16, 17, 18, 19, 20, 21, 22, 25, 27, and 31.

31) Draw a direction field for , determine the behavior as , and state how this behavior may depend on the initial value of y at t = 0.

Look at:

It appears that for , all solutions converge toward something like , while for , all solutions diverge to -.