Ordinary Diff Eq Exam Review Solutions 25

Ordinary Diff Eq Exam Review Solutions 25 - 1 Solutions 27...

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Unformatted text preview: 1 Solutions 27 and simplifying gives . The initial condition implies so for . At we get lbs salt. Once the tank is full, the inflow and outflow rates will be equal and the brine in the tank will (in the limit as ) stabilize to the concentration of the incoming brine, i.e., lb/gal. Since the tank holds 100 gal, the total amount present will approach lbs. Thus . 31. input rate: input rate output rate: output rate Let denote the amount of pollutant at time . Since input rate output rate it follows that is a solution of the initial value problem Rewriting this equation in standard form gives the differential equation . The coefficient function is and the integrating factor is . Thus . Integrating and simplifying gives where is the constant of integration. The initial condition implies so (a) (b) When the river is cleaned up at we assume the input concentration is . The amount of pollutant is therefore given by This will reduce by when . We solve the equation for and get . Similarly, the pollutant will reduce by when . (c) Letting Lake Erie: Lake Ontario: and be given as stated for each lake gives: years, years, years. years 32. Let and denote the amount of salt in Tank 1 and Tank 2, respectively, at time . L g g input rate for Tank 1: input rate . min L min L g g output rate for Tank 1: output rate . min L m The initial value problem for Tank 1 is thus: Simplifying and putting this equation in standard form gives . The integrating factor is . Thus . ...
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