L03 - Review Dierential equations Classication Complete...

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Review Differential equations, Classification, Com- plete Solution (Reduction to Integration) of FOLODE, Existence and Uniqueness Theorem for the general FOODE, Sepa- rable FOODE.
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Example: The FOODE dy dx = x 2 1 - y 2 can be rewritten as x 2 dx +( y 2 - 1) dy = 0 and x 3 / 3+ y 3 / 3 - y = c . This is a cubic equation for y , so all we can say is that the integral curves y ( x ) of the solutions all satify x 3 / 3 + y 3 ( x ) / 3 - y ( x ) = c . c a constant. 2
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Applications of FOODE Example: A tank contains G gallons of water in which are dissolved initially S 0 pounds of salt. Water of salinity s [lb/gal] is added at the rate r [gal/min]. The so- lution leaves the tank at the same rate r . flow-rate-in: r [gal/min] salinity of inflow: s [lb/gal] flow-rate-out: r [gal/min S 0 lb Salt at time 0 G gallons saline solution What is the amount of salt in the tank at any given time t ? 3
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Applications of FOODE Example: A tank contains G gallons of water in which are dissolved initially S 0 pounds of salt. Water of salinity s [lb/gal] is added at the rate r [gal/min]. The so-
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