M427K Handout: Examples of Basic DiffEqs and First-OrderLinear ODEsSalman ButtJanuary 24, 2006This handout provides examples of some introductory problems in differential equations, as wellas specific first-order linear differential equations.Example 1: A certain drug is being administered intravenously to a hospital patient. Fluidcontaining 5 mg/cm3of the drug enters the patient’s blo odstream at a rate of 100 cm3/hr. Thedrug is absorbed by body tissues or otherwise leaves the bloodstream at a rate proportional to theamount present, with a rate constant of 0.4 (hr)−1.(a) Assuming the drug is always uniformly distributed throughout the bloo dstream, write adifferential equation for the amount of the drug that is present in the bloodstream at any time.(b) How much of the drug is present in the bloodstream after a long time ?Example 2: Determine the values of r for which the given diffeq has solutions of the formy = trfor t > 0:t2y00− 4ty0+ 4y = 0Example 3: Find the general solution to the differential equationy0+ y = 5 sin 2t1Example 4: Find the general solution to the differential equation(1 + x3)y0− 3x2y = x2+ x5Example 5: Find the general solution to the differential equation(1 + t2)y0+ 4ty = (1 +
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