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
Want to see the full answer?
Check out a sample textbook solutionChapter 4 Solutions
Differential Equations: An Introduction to Modern Methods and Applications
Additional Math Textbook Solutions
Thinking Mathematically (7th Edition)
Fundamentals of Differential Equations and Boundary Value Problems
Mathematical Methods in the Physical Sciences
Calculus for Business, Economics, Life Sciences, and Social Sciences (13th Edition)
Mathematics with Applications In the Management, Natural, and Social Sciences (12th Edition)
Mathematical Ideas (13th Edition) - Standalone book
- (i) Find the solution set of the following equations: 2 log, x-3log, y=7 and log, x-2log, y = 4 (ii).Given that log, (y-1)+ log, 2. = z and log, (y+1)+ log, x z-1, Show that y 1+8 and find the possible value(s) of y and x when z=1arrow_forwardDraw the phase portraits of the following linear systems and justify the choice of the direction of trajectories. (4.1) * = ( 13 1³ ) *. X, -3 (4.2) = *-(34)x X.arrow_forward(i) Find the solution set of the following equations: 2log; x- 3log, y =7 and log, x-2log; y = 4 (ii).Given that log, (y-1)+log.= |= z and log, (y +1)+log, x = z-1, 4 Show that y? =1+8 and find the possible value(s) of y and x when z=1 (iii). Two executives in cities 400 miles apart drive to a business meeting at a location on the line between their cities. They meet after 4 hours. Find the speed of each car if one car travels 20 miles per hour faster than the other.arrow_forward
- 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. 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_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. 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 + 551 (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) = 0.000004x3 − 0.03x2 + 400x + 80,000 where C(x) denotes the total cost incurred in producing x sets. Find the level of production that will yield a maximum profit for the manufacturer. Hint: Use the quadratic formula. (Round your answer to the nearest whole number.) unitsarrow_forward
- 2. 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_forwardFigure 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_forwardConsider a problem of two connected tanks T1 and T2 containing 300 gallons of water each. Initially, Tank 1 contains 501b of salt while tank 2 contains 100 lb of salt. The liquid circulate between the two tanks at the rate of 10 gallons/min. Tank T1 Tank T2 300 gal 300 gal The amount of salt in T1 and T2 represented by Y1(t) and Y2(t) changes with time t due to flow of liquid and continuous stirring. 1. Calculate the amount of salt in each tank after 15 min. 2. How long will it take to get equal amount of salt in each tank.arrow_forward
- Linear Algebra: A Modern IntroductionAlgebraISBN:9781285463247Author:David PoolePublisher:Cengage Learning