Solve differential equation using initial concentration

Question

The concentration of a drug in a body can be modelled by $\frac{dC}{dt}=-\frac{C}{T}$

Where $T>0$ is a constant that determines how quick the concentration decays within the body. If the inital concentration of the drug is $C_0$, find the solution to this equation.

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Answer 1

We can rewrite it as, $$\frac{dC}C = -\frac{dt}T$$ On integrating, $$\ln C = -\frac tT +k$$

At $t=0$, $C= C_0\implies k = \ln C_0$

Thus, $$\ln C - \ln C_0 = \ln\frac{C}{C_0}=-\frac tT\implies C = C_0e^{-t/T}$$

Answer 2

$\int_{C_0}^C\frac{dC}{C}=-\int_0^t\frac{dt}{T}$

$ln\frac{C}{C_0}=-\frac{t}{T}$

Therefore, $C=C_0e^\frac{-t}{T}$