In each of Problems 1 through 26:
(a) Find the general solution in terms of real functions.
(b) From the roots of the characteristics equation, determine whether each critical point of the corresponding dynamical system is asymptotically stable, stable, or unstable, and classify it as to type.
(c) Use the general solution obtained in part (a) to find a two parameter family of trajectories
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- 13. Find the equation of the orthogonal trajectories of the system of parabolas y² = 2x + 2. a. y = ce-x b. y = cex c. y = ce²x d. y = ce-2xarrow_forwardThe weekly demand for the Pulsar 40-in. high-definition television is given by the demand equation p = −0.05x + 600 (0 ≤ x ≤ 12, 000) where p denotes the wholesale unit price in dollars and x denotes the quantity demanded. The weekly total cost function associated with manufacturing these sets is given by C(x) = 0.000002x 3 − 0.03x 2 + 250x + 80, 000 where C(x) denotes the total cost incurred in producing x sets. 4. Give the profit that corresponds to your production level from part (3). If necessary, round profit to the nearest cent.arrow_forwardThe weekly demand for the Pulsar 40-in. high-definition television is given by the demand equation p = −0.05x + 600 (0 ≤ x ≤ 12, 000) where p denotes the wholesale unit price in dollars and x denotes the quantity demanded. The weekly total cost function associated with manufacturing these sets is given by C(x) = 0.000002x 3 − 0.03x 2 + 250x + 80, 000 where C(x) denotes the total cost incurred in producing x sets. 2. Find a profit function P(x) that gives Pulsar’s weekly revenue in dollars when x sets are sold. Write that function out explicitly.arrow_forward
- The weekly demand for the Pulsar 40-in. high-definition television is given by the demand equation p = −0.05x + 600 (0 ≤ x ≤ 12, 000) where p denotes the wholesale unit price in dollars and x denotes the quantity demanded. The weekly total cost function associated with manufacturing these sets is given by C(x) = 0.000002x 3 − 0.03x 2 + 250x + 80, 000 where C(x) denotes the total cost incurred in producing x sets. 3. Determine the level of production x (with 0 ≤ x ≤ 12, 000) that will yield the maximum profit for the manufacturer. Make sure to show that this x corresponds to an absolute maximum of profit using a strategy we have discussed. You may find it useful to use the quadratic formula. If necessary, round units to the nearest whole television set at the end of your calculations.arrow_forwardThe weekly demand for the Pulsar 40-in. high-definition television is given by the demand equation p = −0.05x + 600 (0 ≤ x ≤ 12, 000) where p denotes the wholesale unit price in dollars and x denotes the quantity demanded. The weekly total cost function associated with manufacturing these sets is given by C(x) = 0.000002x 3 − 0.03x 2 + 250x + 80, 000 where C(x) denotes the total cost incurred in producing x sets. 1. Find a revenue function R(x) that gives Pulsar’s weekly revenue in dollars when x sets are sold. Write that function out explicitly. 2. Find a profit function P(x) that gives Pulsar’s weekly revenue in dollars when x sets are sold. Write that function out explicitly. 3. Determine the level of production x (with 0 ≤ x ≤ 12, 000) that will yield the maximum profit for the manufacturer. Make sure to show that this x corresponds to an absolute maximum of profit using a strategy we have discussed. You may find it useful to use the quadratic formula. If necessary, round units to…arrow_forward2. A company that manufactures pet toys calculates that its costs and revenue can be modeled by the equations C = 15,000 + 1.95x and R = 500x − x2 25 where x is in the number of toys produced in 1 week. Production during one particular week is 5000 toys and is increasing at a rate of 350 toys per week. Find the rate at which the cost, revenue, and profit are changing. a) cost dC dt = dollars/week (b) revenue dR dt = dollars/weekarrow_forward
- Figure below shows three reactors linked by pipes. As indicated, the rate of transfer of chemicals through each pipe is equal to a flow rate (Q, with units of cubic meters per second) multiplied by the concentration of the reactor from which the flow originates (c, with units of milligrams per cubic meter). If the system is at a steady-state, the transfer into each reactor will balance the transfer out. Develop mass-balance equations for the reactors and solve the three simultaneous linear algebraic equations for their concentrations using gauss_partial.m and lu_decomp.marrow_forwardFind the general solution in terms of real functions. (b) From the roots of the characteristic equation, determine whether each critical point of the corresponding dynamical system is asymptotically stable, stable, or unstable, and classify it as to type. (c) Use the general solution obtained in part (a) to find a two-parameter family of trajectories x=x1i+x2j=yi+y′j of the corresponding dynamical system. Then sketch by hand, or use a computer, to draw a phase portrait, including any straight-line orbits, from this family of trajectories.arrow_forwardInitially, two large tanks A and B each hold 100 gallons of brine. The well-stirred liquid is pumped between the tanks as shown in the figure below. Use the information given in the figure to construct a mathematical model for the number of pounds of salt x1(t) and x2(t) at time t, measured in minutes, in tanks A and B, respectively.arrow_forward
- Linear Algebra: A Modern IntroductionAlgebraISBN:9781285463247Author:David PoolePublisher:Cengage Learning