# Modelling Differential Equations

Here we will look at setting up DEs for various applications.

## Mixtures

Suppose a tank contains a mixture with some initial concentration. There can be flows coming in and out of the rank at different rates and concentrations. Overall, the change in concentation is:

$$\cfrac{dA}{dt} = rate_{in} - rate_{out}$$

Chocolate containing $$0.04 \frac{kg}{L}$$ of almonds enters a tank at a rate of $$3 \frac{L}{hr}$$. The mixture is well mixed and drains at a rate of $$5 \frac{L}{hr}$$. The tank originally has $$100 L$$ containing $$1 \; kg$$ of almonds. Write a DE and describe how to solve for $$A(t)$$ the amount of almonds at time $$t$$.

A radioactive element decays at a constant rate proportional to the nunmber of remaining nuclides:

$$\frac{dN}{dt} = -kN$$

This is a linear DE that is separable. It can be solved by direct integration:

$$N = N_0 e^{-kt}$$

Where $$N_0$$ is the initial amount in grams of number of nuclides.

## Electric Circuits

A series circuit with a resistor and inductor has the following DE:

$$L \cfrac{di}{dt} + Ri = E$$

This is a linear DE that is also separable:

$$\cfrac{di}{E - Ri} = \cfrac{dt}{L}$$

$$\ln {(E - Ri)} = c_1 - \cfrac{Rt}{L}$$

$$E - Ri = c_2e^{\cfrac{-Rt}{L}}$$

$$Ri = E - c_2e^{\cfrac{-Rt}{L}}$$

$$i(t) = \cfrac{E}{R} + C e^{\cfrac{-Rt}{L}}$$