Differential equations have been used extensively in the study of drug dissolution for patients given oral medication. One such equation is the weibull equation for the concentration c(t) of the drug:
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Differential equations have been used extensively in the study of drug dissolution for patients given oral medications. The three simplest equations used are the zero-order kinetic equation, the Noyes-Whitney equation, and the Weibull equation. All assume that the initial concentration is zero but make different assumptions about how the concentration increases over time during the dissolution of the medication. The Noyes-Whitney equation for the dynamics of the drug concentration is dcdt=k(cs−c)\frac{d c}{d t}=k\left(c_{s}-c\right)dtdc=k(cs−c) where k and csc_scs are positive constants. Is this differential equation pure-time, autonomous, or nonautonomous? State in words what this differential equation says about how drug dissolution occurs. Verify that c=cs(1−e−kt)c=c_{s}\left(1-e^{-k t}\right)c=cs(1−e−kt) is the solution to this equation for the initial condition c(0) = 0.