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Differential Equations as Mathematical Models Population Dynamics Animal Population The number of field mice in a certain dP P pasture is given by the function 200-10t, where t is measured in years. Determine a dt DE governing a population of owls that feed on the mice if the rate at which the owl population grows is proportional to the difference between the number of owls and field mice at time t. dP kP dP dt k P (200 10t ) dt Newton’s Law of Cooling Water Temperature Water is heated to the boiling point dT temperature of 100ºC. The water is then T Tm removed from the heat and kept in a room which is at a constant temperature dt of 22.5ºC. After 3 minutes the water temperature is 90ºC. Find the water temperature after 9 minutes. When will the water temperature be 50ºC? dT k T Tm dT k T 22.5 , T (3) 90 dt dt T (9) ?, T (?) 50. Spread of Disease Suppose a student carrying a flu virus returns to an isolated college campus of dNi 1000 students. Determine a differential Ni Nu equation governing the number of people Ni who have contracted the flu if the rate dt at which the flu spreads is proportional to the number of interactions between the number of students that have the flu, Ni, and those that do not have it yet Nu. dN i kN i N u dN i kN i 1000 N i dt dt Chemical Reactions Two chemicals, A and B, react to form another C. It is found that the rate at dx x x which C is formed varies as the product of the instantaneous amounts dt of chemicals A and B present. The formation requires 2lb of A for each pound of B. If 10 lb of A and 20 lb of B are present initially, and if 6 lb of C are formed in 20 min, find the amount of chemical C at any time. 2 lb of A dx for k x x dC dt k 10 23x 20 3 , x dt C (0) 0, C (20) 6 Mixtures A tank has 10 gal brine having 2 lb of dissolved salt. Brine with 1.5 lb of salt per gallon enters at 3 gal/min, and the well- stirred mixture leaves at 4 gal/min. Find the amount of salt in the tank at any time. is is dA Rin Rout dt dA 4A 3(1.5) dt 10 t dA 4A 4.5 , A(0) 2 dt 10 t Torricelli’s Law V Ah s A right-circular cylindrical tank leaks water out of a circular hole at its bottom. If friction and contraction of the water stream near the hole reduce the volume dV Ah ds Ahv dt dt of the water leaving the tank per second to: cA 2 gh , where c 0.6 h Determine a DE for the height, h, of the dV water at time t if the radius of the Ah 2 gh cylinder is 2 ft and that of the hole is 2 dt in. Assume g = 32 ft/sec. dh Aw 2 gh Ah dh dt Aw Ah 2 gh dh dt 24 64h 18 4 2 h dt Series Circuits An inductor of 0.5 henry is connected dI L RI E in series with a resistor of 6 ohms, a capacitor of 0.02 farad, a generator dt having alternating voltage given by 24 sin(10t), t 0, and a switch. Find the charge and current at time t if the dq 1 charge on the capacitor is zero when R qE the switch is closed at t=0. dt C q 12q 100q 48sin(10t ) 2 d q dq 1 L 2 R qE dt dt C Falling Bodies A sky diver with a parachute falls from rest. Let the combined weight of the sky diver and parachute be 200 lb. If the parachute encounters an air dv resistance equal to 1.5, where m mg kv is the speed at any instant during the fall, that she falls vertically dt downward, and that the parachute is already open when the jump takes place, describe the ensuing dv motion. 25 800 6v, v(0) 0 dt x 0.24 x 32 0, x(0) 0, x(0) 0 Newton’s 2nd Law of Motion A uniform chain of length L and linear density lies in a heap on an edge of a smooth table and starts sliding over the edge. Get a DE that governs the motion of F d ( vm ) dt v dm m dv dt dt the chain during the time it is sliding over the edge? m y, dm dt ;v dy dt dy dt . mg v dm dt m dv dt F dy 2 dt y d2y dt 2 mg yg y yy 2 yg y yy 2 Miscellaneous Models--1 Hanging Cable-general Let a cable or rope be hung from two points, not necessarily at the same level. Assume that the cable is flexible so T sin W ( x), T cos H that it curves due to its load (its own weight, tan dy dx WHx ) ( external forces, or a combination of the two). Determine a DE that governs the shape of d2y dx 2 1 dW H dx the cable. Miscellaneous Models--2 Suspension Bridge d2y dx 2 1 dW H dx A flexible cable of small (negligible) weight supports a uniform bridge. Determine the shape of the cable. The fact that the bridge is uniform d2y dx 2 , H tellsdw that us dx is a constant, say . Then the DE that models this y (0) b, y(0) 0 problem is as is seen on the left. Miscellaneous Models--3 Hanging Cable dW ds dW dx dx ds Let a flexible cable having a constant dW dx dx ds density, say , hangs between two fixed points. Determine the shape of the cable, if we assume the only 2 ds 1 dy , force acting on the cable is its own dx dx weight. d2y 1 dW 2 dW dx 1 dy dx dx 2 H dx Here we first dW note that 2 ds d2y dx 2 H 1 dy dx , where s, is the length of the rope. y (0) b, y(0) 0 Miscellaneous Models --4 Shape of a Reflector DPR PFE FP0; Find the shape of a reflector that reflects all light rays coming POG 2 PFE; parallel to a fixed axis to a single point. tan( ) dy dx y; y x tan(2 ) 1tan 2 1 y2 2 tan 2y 2 xy y 1 y 2

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posted: | 3/23/2012 |

language: | English |

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