Consider the differential equation dy/dt = ay b.(a) Find

Solve each of the following initial value problems and plot the solutions for several values of y0. Then describe in a few words how the solutions resemble, and differ from, each other. (a) dy/dt = y + 5, y(0) = y0 (b) dy/dt = 2y + 5, y(0) = y0 (c) dy/dt = 2y + 10, y(0) = y0

Follow the instructions for Problem 1 for the following initial value problems: (a) dy/dt = y 5, y(0) = y0 (b) dy/dt = 2y 5, y(0) = y0 (c) dy/dt = 2y 10, y(0) = y0

Consider the differential equation dy/dt = ay + b, where both a and b are positive numbers. (a) Find the general solution of the differential equation. (b) Sketch the solution for several different initial conditions. (c) Describe how the solutions change under each of the following conditions: i. a increases. ii. b increases. iii. Both a and b increase, but the ratio b/a remains the same.

Consider the differential equation dy/dt = ay b. (a) Find the equilibrium solution ye. (b) Let Y(t) = y ye; thus Y(t) is the deviation from the equilibrium solution. Find the differential equation satisfied by Y(t).

Undetermined Coefficients. Here is an alternative way to solve the equation dy/dt = ay b. (i) (a) Solve the simpler equation dy/dt = ay. (ii) Call the solution y1(t). (b) Observe that the only difference between Eqs. (i) and (ii) is the constant b in Eq. (i). Therefore, it may seem reasonable to assume that the solutions of these two equations also differ only by a constant. Test this assumption by trying to find a constant k such that y = y1(t) + k is a solution of Eq. (i). (c) Compare your solution from part (b) with the solution given in the text in Eq. (17). Note: This method can also be used in some cases in which the constant b is replaced by a function g(t). It depends on whether you can guess the general form that the solution is likely to take. This method is described in detail in Section 3.5 in connection with second order equations

Use the method of Problem 5 to solve the equation dy/dt = ay + b.

The field mouse population in Example 1 satisfies the differential equation dp/dt = 0.5p 450. (a) Find the time at which the population becomes extinct if p(0) = 850. (b) Find the time of extinction if p(0) = p0, where 0 < p0 < 900. (c) Find the initial population p0 if the population is to become extinct in 1 year.

Consider a population p of field mice that grows at a rate proportional to the current population, so that dp/dt = rp. (a) Find the rate constant r if the population doubles in 30 days. (b) Find r if the population doubles in N days.

The falling object in Example 2 satisfies the initial value problem dv/dt = 9.8 (v/5), v(0) = 0. (a) Find the time that must elapse for the object to reach 98% of its limiting velocity. (b) How far does the object fall in the time found in part (a)?

Modify Example 2 so that the falling object experiences no air resistance. (a) Write down the modified initial value problem. (b) Determine how long it takes the object to reach the ground. (c) Determine its velocity at the time of impact.

Consider the falling object of mass 10 kg in Example 2, but assume now that the drag force is proportional to the square of the velocity. (a) If the limiting velocity is 49 m/s (the same as in Example 2), show that the equation of motion can be written as dv/dt = [(49) 2 v2 ]/245. Also see Problem 25 of Section 1.1. (b) If v(0) = 0, find an expression for v(t) at any time. (c) Plot your solution from part (b) and the solution (26) from Example 2 on the same axes. (d) Based on your plots in part (c), compare the effect of a quadratic drag force with that of a linear drag force. (e) Find the distance x(t) that the object falls in time t. (f) Find the time T it takes the object to fall 300 m.

A radioactive material,such as the isotope thorium-234,disintegrates at a rate proportional to the amount currently present. If Q(t)is the amount present at time t, then dQ/dt = rQ, where r > 0 is the decay rate. (a) If 100 mg of thorium-234 decays to 82.04 mg in 1 week, determine the decay rate r. (b) Find an expression for the amount of thorium-234 present at any time t. (c) Find the time required for the thorium-234 to decay to one-half its original amount.

The half-life of a radioactive material is the time required for an amount of this material to decay to one-half its original value. Show that for any radioactive material that decays according to the equation Q= rQ, the half-life and the decay rate r satisfy the equation r = ln 2.

Radium-226 has a half-life of 1620 years. Find the time period during which a given amount of this material is reduced by one-quarter.

According to Newtons law of cooling (see Problem 23 of Section 1.1), the temperature u(t) of an object satisfies the differential equation du dt = k(u T), where T is the constant ambient temperature and k is a positive constant. Suppose that the initial temperature of the object is u(0) = u0. (a) Find the temperature of the object at any time. (b) Let be the time at which the initial temperature difference u0 T has been reduced by half. Find the relation between k and .

Suppose that a building loses heat in accordance with Newtons law of cooling (see Problem 15) and that the rate constantk has the value 0.15 h1.Assume that the interior temperature is 70F when the heating system fails. If the external temperature is 10F, how long will it take for the interior temperature to fall to 32F?

Consider an electric circuit containing a capacitor, resistor, and battery; see Figure 1.2.3. The charge Q(t) on the capacitor satisfies the equation5 RdQ dt + Q C = V, where R is the resistance, C is the capacitance, and V is the constant voltage supplied by the battery. (a) If Q(0) = 0, find Q(t) at any time t, and sketch the graph of Q versus t. (b) Find the limiting value QL that Q(t) approaches after a long time. (c) Suppose that Q(t1) = QL and that at time t = t1 the battery is removed and the circuit is closed again. Find Q(t) for t > t1 and sketch its graph.

A pond containing 1,000,000 gal of water is initially free of a certain undesirable chemical (see Problem 21 of Section 1.1).Water containing 0.01 g/gal of the chemical flows into the pond at a rate of 300 gal/h, and water also flows out of the pond at the same rate. Assume that the chemical is uniformly distributed throughout the pond. (a) Let Q(t) be the amount of the chemical in the pond at time t. Write down an initial value problem for Q(t). (b) Solve the problem in part (a) for Q(t). How much chemical is in the pond after 1 year? (c) At the end of 1 year the source of the chemical in the pond is removed; thereafter pure water flows into the pond, and the mixture flows out at the same rate as before.Write down the initial value problem that describes this new situation. (d) Solve the initial value problem in part (c). How much chemical remains in the pond after 1 additional year (2 years from the beginning of the problem)? (e) How long does it take for Q(t) to be reduced to 10 g? (f) Plot Q(t) versus t for 3 years.

Your swimming pool containing 60,000 gal of water has been contaminated by 5 kg of a nontoxic dye that leaves a swimmers skin an unattractive green. The pools filtering system can take water from the pool, remove the dye, and return the water to the pool at a flow rate of 200 gal/min. (a) Write down the initial value problem for the filtering process; let q(t) be the amount of dye in the pool at any time t. (b) Solve the problem in part (a). (c) You have invited several dozen friends to a pool party that is scheduled to begin in 4 h. You have also determined that the effect of the dye is imperceptible if its concentration is less than 0.02 g/gal. Is your filtering system capable of reducing the dye concentration to this level within 4 h? (d) Find the time T at which the concentration of dye first reaches the value 0.02 g/gal. (e) Find the flow rate that is sufficient to achieve the concentration 0.02 g/gal within 4 h.