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The pathway for a binary electrical signal between gates in an integrated circuit can be modeled as an
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Chapter 3 Solutions
Fundamentals of Differential Equations and Boundary Value Problems
- 2713 - 1 dt 3.arrow_forwardTwo tanks are connected as in Figure 1.6. Tank 1 initially contains 20 pounds of salt dissolved in 100 gallons of brine. Tank 2 initially contains 150 gallons of brine in which 90 pounds of salt are dissolved. At time zero, a brine solution containing 1/2 pound of salt per gallon is added to tank I at the rate of 5 gallons per minute, Tank 1 has an output that discharges brine into tank 2 at the rate of 5 gallons per minute, and tank 2 also has an output of 5 gallons per minute. Deter- mine the amount of salt in each tank at any time. Also, determine when the concentration of salt in tank 2 is a minimum and how much salt is in the tank at that time. Hint: Solve for the amount of salt in tank 1 at time t and use this solution to help determine the amount in tank 2. S prl min: 12 Ih gel S gil tain Tauk pal minarrow_forwardConsider the two tanks shown in the figure below. Assume that tank A contains 50 gallons of water in which 25 pounds of salt is dissolved. Suppose tank B contains 50 gallons of pure water. Liquid is pumped into and out of the tanks as indicated in the figure; the mixture exchanged between the two tanks and the liquid pumped out of tank B are assumed to be well stirred. We wish to construct a mathematical model that describes the number of pounds x₁(t) and x₂(t) of salt in tanks A and B, respectively, at time t. dx₁ dt dx₂ dt mixture 4 gal/min This system is described by the system of equations 1 50 2 2 25 2 dx₁ dt dx2 dt = = pure water 3 gal/min = 2 25 2 110 25 A + 1 1 mixture 1 gal/min B with initial conditions x₁(0) = 25, x₂(0) = 0 (see (3) and the surrounding discussion on mixtures on page 107). What is the system of differential equations if, instead of pure water, a brine solution containing 4 pounds of salt per gallon is pumped into tank A? mixture 3 gal/minarrow_forward
- Given an RL circuit with the equation dl L + RI = 900, dt where L is the inductance, R is the resistance and 900 is the voltage source. Show that the solution of the above equation is 900 I(t) = + Ce. Rarrow_forwardQUESTION 4 4.1 A motor company manufacture and sell cars and motorbikes. The cost of manufacturing x motorbikes and y cars is given by C(x, y) = 100x² +100xy + 400y. Each motorbike is sold for N$36 000-00 and each car is sold for N$180 000-00. Use Cramer's rule to determine the number of motorbikes and the number of cars that should be manufactured and sold for a maximum profit II and determine the maximum profit IImax : 4.2 Use the Jacobian to test for functional dependence between the cost and the revenue functions in 4.1. One of the stationary points of the function f(x, y) = x* +y - 2x +4xy-2y is (2,-V2). 4.3 %3D Use the Hessian to test whether the given point is a maximum, minimum or a saddle point.arrow_forwardSuppose that each year, 80% of residents living in state A stay in state A, while 20% of them move to state B. Also, 70% of the residents living in state B stay in state B, while 30% of them move to state A. Further, assume at t = 0 years, there are 2 million residents living in state A and 5 million residents living in state B, and let æn and yn represent the populations (in millions) of the states A and B, respectively, at time t = n years. Given that Xn a 2 Yn d find the value of the number c.arrow_forward
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