Modeling with differential equations

In this section, we will try to apply differential equations to real life situations. For each question we will look how to set up the differential equation. Afterwards, we will find the general solution and use the initial condition to find the particular solution. Depending on the question, we will even look at behaviours of the differential equation and see if it is applicable to real life situations. For example, one can notice that integrating the area of a sphere actually gives the volume of a sphere!

We will be learning how to create a differential equation out of the word problem, and then find the general and particular solutions. We will then take a look at the behaviour of the general solution to find results we need to answer the questions.

It may be convenient to use the following formula when modelling differential equations related to proportions:

$\frac{dy}{dt}=kM$

Where:
1. $\frac{dy}{dt}$ is the rate of change of $y$
2. $k$ is a constant
3. $M$ is the equation that models the problem

There are many applications to first-order differential equations. Some situations that can give rise to first order differential equations are:
• Radioactive Decay
• Population Dynamics (growth or decline)

Exponential Model:
$\frac{dP}{dt}=KP$
$P=Ce^{Kt}$

Logistic Model:
$\frac{dP}{dt}=KP(1-\frac{P}{M})$
$P= \frac{M}{1+Ce^{-kt}}$ $C= \frac{M-P_0}{P_0}$

• Newton's Law of Cooling
$\frac{dT}{dt}=-K(T-T_a)$
$|T-T_a |=Ce^{-kt}$
If $T \geq T_a$ (the object is more hot), then $T=Ce^{-kt}+T_a$
If $T$ < $T_a$ (the object is cooler), then $T=T_a-Ce^{-kt}$