Modelling Differential Equations

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


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 \(t\).

Radioactive Decay

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:

\( \cfrac{di}{E - Ri} = \cfrac{dt}{L}\)

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

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

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

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

Try these questions: