# Modeling with differential equations

### 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!

#### Lessons

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:
• 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 $T=T_a-Ce^{-kt}$
• 1.
a)
What sort of problems can be solved by first-order differential equations?

b)
Modeling Population Dynamics

c)
Logistic Modeling

d)
Newton’s Law of Cooling

• 2.
Modeling and Solving Differential Equations
The rate of change of the volume of a spherical balloon in terms of radius is equal to the surface area of the balloon itself. Find the general solution of the differential equation.

• 3.

You are working in a factory that converts liquid X to liquid Y at a rate that is proportional to the cubic of the amount of X. Initially, you start with 100 grams of liquid X. After 2 hours, you only have 30 grams of liquid X remaining. How much of liquid X is remaining after 4 hours?

• 4.
Solving and Examining the Behaviour of the General Solution
You are studying for a math test that is out of 100. Assume that if you do not study, then you will get 30% on your test. The mark on your math test is related to how much you study according to the model

$S\frac{dS}{dt}=1$

Where $S$ is the mark on your math test out of 100 and $t$ is the amount of hours studied.

a) Solve this differential equation
b) How much do you have to study in order to get 100 on your math test?
c) Is it possible to get over 100 on your math test?

• 5.
Let $F(t)$ be the number of Facebook friends you have at time $t$. Assume that $F(t)$ satisfies the logistic growth equation:

$\frac{dF}{dt}=0.1F(1-\frac{F}{100})$

a) Find $F(t)$. (do not find the constant)
b) Find $\lim$t →$\infty$ $F(t)$. What does this result mean?

• 6.
Population Dynamics
A population of Nudibranchs grow at a rate that is proportional to their current population. Without any outside factors the population will double every month.
If the population is originally 100 Nudibranchs, how many Nudibranchs will there be after 3 months?

• 7.
Logistic Modeling
A population of Nudibranchs grow at a rate that is proportional to their current population. Without any outside factors the population will double every month. However the ecosystems the Nudibranchs are living in can only support 15,000 Nudibranchs. If the population is originally 100 Nudibranchs, how many Nudibranchs will there be after 3 months?

• 8.
Newton’s Law of Cooling
Dr. Daniel Pierce is investigating a crime scene. A corpse was found and the temperature was recorded to be 26°C with the surrounding room temperature being 22°C. The temperature of the corpse was measured two hours later and was found to be 24°C. Assuming that prior to being murdered the person was 37°C, how long ago was the murder committed?