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Unformatted text preview: Notes for Day : . : Introduction to Mathematica Mode s In the last section, we discussed methods for solving the rst order linear homogeneous di erential equation and the rst order linear nonhomogeneous di erential equation . Today, well discuss some important types of applications that can be solved using one of these types of di erential equations. Mixing Prob ems In mixing problems, we are modelling the change in some quantity y over time, when that quantity is changing due to in ow and out ow, possibly occurring simultaneously. In the standard example of this type of problem, salt water ows into a tank, and is mixed with whatever uid is already in the tank, and then the mixture ows out of the tank. We make the explicit assumption that mixing occurs instantaneouslythis assumption simpli es our calculations; furthermore, it does not introduce an unreasonable amount of error if the speed of mixing is very quick compared to the rate of ow. When setting up these problems, keep in mind the following: e amount of salt owing in or out of the system is directly proportional to the amount of water owing in or out of the system. e actual proportion is determined by the concentration. e rate of change of the amount of salt inside the tank must be equal to the di erence between the rate at which salt ows in, and the rate at which salt ows out....
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