Introduction

Welcome to this comprehensive guide on modeling situations with differential equations! This topic introduces you to one of the most powerful tools in mathematics for describing how quantities change over time or with respect to other variables. Unlike algebraic equations that relate variables directly, differential equations relate a function to its derivatives, allowing us to model dynamic processes and predict future behavior based on rates of change.

The essential knowledge tells us that differential equations relate a function of an independent variable and the function's derivatives. This means we can describe situations where we know how fast something is changing (the derivative) and use that information to find the original function. This approach is fundamental to modeling real-world phenomena in physics, biology, economics, and engineering.

Understanding Differential Equations

A differential equation is an equation that contains a function and one or more of its derivatives. The general form can be written as:

$F(x,y,y^{\prime },y^{\prime \prime },\dots ,y^{(n)})=0$

where $y$ is the unknown function of $x$, and $y^{\prime }$, $y^{\prime \prime }$, etc., are its derivatives.

In this course, we focus primarily on first-order differential equations, which involve only the first derivative $\frac{dy}{dx}$. These equations have the general form:

$\frac{dy}{dx}=f(x,y)$

The order of a differential equation is determined by the highest derivative present. For example:

Translating Verbal Statements to Differential Equations

The key skill in this topic is interpreting verbal descriptions and converting them into mathematical differential equations. Let's examine common patterns:

Rate of Change Language

Example 1: Population Growth

"The rate of change of a population is proportional to the current population."

Key phrases to identify:

This translates to: $\frac{dP}{dt}=kP$, where $k$ is the proportionality constant.

Example 2: Cooling Process

"The rate at which the temperature of an object changes is proportional to the difference between the object's temperature and the ambient temperature."

Breaking this down:

This gives us Newton's Law of Cooling: $\frac{dT}{dt}=-k(T-T_{ambient})$

Note the negative sign because objects cool down when they're warmer than their surroundings.

Exponential Growth and Decay Models

Example 3: Radioactive Decay

"A radioactive substance decays at a rate proportional to the amount present."

This translates to: $\frac{dN}{dt}=-kN$, where $k>0$ represents the decay constant.

Example 4: Compound Interest

"Money in an account earns interest continuously at a rate proportional to the current balance."

This gives us: $\frac{dB}{dt}=rB$, where $r$ is the interest rate.

Logistic Growth Models

Example 5: Limited Population Growth

"A population grows at a rate proportional to both the current population and the remaining capacity for growth."

This leads to the logistic equation: $\frac{dP}{dt}=kP(M-P)$

This can also be written as: $\frac{dP}{dt}=kP\left(1-\frac{P}{M}\right)$

Mixing Problems

Example 6: Salt Concentration

"Salt water is flowing into and out of a tank. The rate of change of salt in the tank equals the rate of salt entering minus the rate of salt leaving."

Let $S(t)$ = amount of salt at time $t$, $V(t)$ = volume of solution at time $t$

This gives us: $\frac{dS}{dt}=c_{in}\cdot r_{in}-\frac{S(t)}{V(t)}\cdot r_{out}$

Identifying Key Components

When translating verbal problems, look for these key elements:

Example 7: Drug Concentration

"The rate at which a drug is eliminated from the bloodstream is proportional to the concentration of the drug."

Multiple Factors Affecting Rate

Example 8: Predator-Prey Model

"The rate of change of the prey population depends on natural growth of prey and losses due to predation."

If $x$ = prey population and $y$ = predator population:

This gives us: $\frac{dx}{dt}=ax-bxy$

Example 9: Spread of Disease

"The rate of spread of a disease is proportional to the number of infected individuals and the number of susceptible individuals."

If $I$ = infected, $S$ = susceptible: $\frac{dI}{dt}=kSI$

Setting Up Initial Value Problems

Many real-world problems include initial conditions. These create initial value problems (IVPs).

Example 10: Population with Initial Condition

"A population grows at a rate proportional to its size. Initially, there are 1000 individuals."

This gives us the IVP: $\frac{dP}{dt}=kP,P(0)=1000$

Example 11: Cooling with Initial Temperature

"A cup of coffee at 90°C is placed in a room at 20°C. The rate of cooling is proportional to the temperature difference."

This gives us: $\frac{dT}{dt}=-k(T-20),T(0)=90$

Recognizing Differential Equation Types

Separable Equations: Can be written as $\frac{dy}{dx}=f(x)g(y)$

Example: $\frac{dy}{dx}=xy$ is separable because $f(x)=x$ and $g(y)=y$.

Linear First-Order Equations: Have the form $\frac{dy}{dx}+P(x)y=Q(x)$

Example: $\frac{dy}{dx}+2xy=x$ is linear.

Autonomous Equations: The right side depends only on $y$, not on $x$

Example: $\frac{dy}{dx}=y^2-4y$ is autonomous.

Common Mathematical Models

Exponential Growth/Decay

Model: $\frac{dy}{dt}=ky$

Applications: Population growth, radioactive decay, compound interest

Logistic Growth

Model: $\frac{dy}{dt}=ky(M-y)$

Applications: Limited population growth, spread of innovations

Newton's Law of Cooling

Model: $\frac{dT}{dt}=-k(T-T_{ambient})$

Applications: Temperature changes, forensic science

Mixing Problems

Model: $\frac{dS}{dt}=\text{rate in}-\text{rate out}$

Applications: Chemical concentrations, pollution levels

Practice Section