0 upstream then a reasonable model for its concentration in the river is p t 8

# 0 upstream then a reasonable model for its

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= 0 upstream, then a reasonable model for its concentration in the river is p ( t ) = 8 e - 0 . 002 t ppm (parts per million) Form the differential equation describing the concentration of pollutant in the lake at any time t and solve it Graph the solution and approximate how long it takes for this lake to have a concentration of 2 ppm Joseph M. Mahaffy, h [email protected] i Lecture Notes – Linear Differential Equations — (32/64)

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Introduction Falling Cat 1 st Order Linear DEs Examples Pollution in a Lake Example 2 Mercury in Fish Modeling Mercury in Fish Example 2: Pollution in a Lake 2 Solution: This model follows the original derivation above with V = 10 5 , f ( t ) = 100 + 60 sin(0 . 0172 t ), and p ( t ) = 8 e - 0 . 002 t , so the DE for the concentration of pollutant is dc ( t ) dt = - f ( t ) V ( c ( t ) - p ( t )) with c (0) = 0 = - (0 . 001 + 0 . 0006 sin(0 . 0172 t )) ( c ( t ) - 8 e - 0 . 002 t ) This requires use of an integrating factor μ ( t ) = e R (0 . 001+0 . 0006 sin(0 . 0172 t )) dt = e 0 . 001 t - 0 . 0349 cos(0 . 0172 t ) so d dt e 0 . 001 t - 0 . 0349 cos(0 . 0172 t ) c ( t ) = (0 . 008+0 . 0048 sin(0 . 0172 t )) e - 0 . 001 t - 0 . 0349 cos(0 . 0172 t ) Joseph M. Mahaffy, h [email protected] i Lecture Notes – Linear Differential Equations — (33/64)
Introduction Falling Cat 1 st Order Linear DEs Examples Pollution in a Lake Example 2 Mercury in Fish Modeling Mercury in Fish Example 2: Pollution in a Lake 3 Solution: From before, d dt e 0 . 001 t - 0 . 0349 cos(0 . 0172 t ) c ( t ) = (0 . 008 + 0 . 0048 sin(0 . 0172 t )) e - 0 . 001 t - 0 . 0349 cos(0 . 0172 t ) The integrating gives e 0 . 001 t - 0 . 0349 cos(0 . 0172 t ) c ( t ) = Z (0 . 008 + 0 . 0048 sin (0 . 0172 t )) e - 0 . 001 t - 0 . 0349 cos(0 . 0172 t ) dt. This last integral cannot be solved, even with Maple . Numerical methods are needed to solve and graph this problem, and our preferred method is MatLab Joseph M. Mahaffy, h [email protected] i Lecture Notes – Linear Differential Equations — (34/64)

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Introduction Falling Cat 1 st Order Linear DEs Examples Pollution in a Lake Example 2 Mercury in Fish Modeling Mercury in Fish Example 2: Pollution in a Lake 4 MatLab Solution: The pollution problem is integrated numerically (ode23). MatLab finds that the pollution exceeds 2 ppm after t = 447 . 4 days . Below shows a graph. (Programs are provided on Lecture page .) 0 200 400 600 800 1000 1200 1400 1600 1800 2000 0 0.5 1 1.5 2 2.5 3 Critical time t days c ( t ) ppm Lake Pollution Joseph M. Mahaffy, h [email protected] i Lecture Notes – Linear Differential Equations — (35/64)
Introduction Falling Cat 1 st Order Linear DEs Examples Pollution in a Lake Example 2 Mercury in Fish Modeling Mercury in Fish Pollution in a Lake: Complications Pollution in a Lake: Complications The above examples for pollution in a lake fail to account for many significant complications There are considerations of irregular variations of pollutant entering, stratification in the lake, and uptake and reentering of the pollutant through interaction with the organisms living in the lake The river will have varying flow rates, and the leeching of the pollutant into river is highly dependent on rainfall, ground water movement, and rate of pollutant introduction Obviously, there are many other complications that would increase the difficulty of analyzing this model Joseph M. Mahaffy, h [email protected] i Lecture Notes – Linear Differential Equations — (36/64)

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Introduction Falling Cat 1 st Order Linear DEs Examples
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