Modelling with Differential Equations 📈

Introduction: Why do changes matter, students?

Differential equations are one of the most powerful tools in calculus because they describe how a quantity changes over time or in response to something else. Instead of only asking what a value is right now, we ask how fast it is changing and what that tells us about the future. That is the heart of modelling with differential equations. For example, a population may grow faster when it is small and slower when food becomes limited. A hot drink cools faster when it is much warmer than the room. A medicine in the bloodstream may break down at a rate related to how much is present. These are all situations where a rule about change can model the real world 🌍.

In this lesson, students, you will learn the main ideas and terminology behind modelling with differential equations, how to build and interpret simple models, and how this topic connects to the rest of calculus. You will also see how IB Mathematics: Analysis and Approaches HL uses differential equations to link derivatives, integration, and real-life reasoning.

What is a differential equation?

A differential equation is an equation that includes a function and one or more of its derivatives. In simple terms, it tells us a relationship between a quantity and its rate of change. For instance, if $y$ represents a quantity and $x$ is the independent variable, then an equation like $\frac{dy}{dx}=ky$ is a differential equation because it involves the derivative $\frac{dy}{dx}$ and the function $y$ itself.

The key idea is that the derivative describes how something changes. If we know how a quantity changes, we can often find the quantity itself by solving the differential equation. This is why differential equations are closely connected to integration: differentiation describes change, and integration helps recover the original function.

There are two main kinds of differential equations you should recognize:

Ordinary differential equations involve derivatives with respect to one independent variable, such as $\frac{dy}{dt}$.
Partial differential equations involve partial derivatives with respect to more than one variable, but these are usually beyond the core focus of IB AA HL modelling.

In IB calculus, the most common models are ordinary differential equations, especially first-order ones.

Building a model from a real situation

To model a real situation, students, you usually begin by identifying the quantities involved and deciding what variable represents each one. Then you think about how the quantity changes. The model does not need to be perfect; it needs to be useful and mathematically justified.

A common modelling pattern is that the rate of change depends on the current amount. This can lead to equations such as:

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

Here, $y$ might represent population, money in a bank account with continuous growth, or bacteria in a culture. The constant $k$ determines whether the quantity grows or decays:

If $k>0$, the quantity increases exponentially.
If $k<0$, the quantity decreases exponentially.

For example, suppose a bacteria population grows at a rate proportional to its current size. If $P(t)$ is the population at time $t$, then:

$$\frac{dP}{dt}=kP$$

This says the bigger the population, the faster it grows. If you also know the initial population, such as $P(0)=P_0$, then you have an initial value problem. This extra information allows a unique solution.

The solution to this model is:

$$P(t)=P_0e^{kt}$$

This formula is a classic exponential growth model. It shows how a differential equation becomes a complete prediction tool once the initial condition is included.

Common modelling ideas in HL calculus

One important modelling idea is the proportionality rule. Many natural processes have rates proportional to the amount present or the amount remaining. For example, cooling often depends on the difference between an object’s temperature and the room temperature. This leads to Newton’s law of cooling:

$$\frac{dT}{dt}=-k(T-T_{\text{room}})$$

Here, $T$ is the object’s temperature, $T_{\text{room}}$ is the surrounding temperature, and $k>0$ is a constant. The negative sign shows the object cools when it is warmer than the room. As $T$ gets closer to $T_{\text{room}}$, the rate of cooling becomes smaller. That matches real life because a cup of tea cools quickly at first and more slowly later ☕.

Another important idea is equilibrium. An equilibrium occurs when the rate of change is zero. For a differential equation like $\frac{dy}{dt}=f(y)$, an equilibrium value $y=c$ satisfies $f(c)=0$. Equilibria help us understand what values the system may settle near.

For example, in a logistic population model:

$$\frac{dP}{dt}=rP\left(1-\frac{P}{K}\right)$$

$P$ is the population, $r$ is the growth rate, and $K$ is the carrying capacity. The term $\left(1-\frac{P}{K}\right)$ reduces growth as $P$ gets close to $K$. When $P$ is small, the population grows almost like $\frac{dP}{dt}=rP$. When $P$ is near $K$, growth slows down. This model is more realistic than unlimited exponential growth because real populations face limited resources.

Solving simple differential equations

IB AA HL expects you to understand how to solve basic separable differential equations. A differential equation is separable if it can be written so that all the $y$-terms are on one side and all the $x$-terms are on the other.

For example:

$$\frac{dy}{dx}=xy$$

We can rewrite this as:

$$\frac{1}{y}dy=x\,dx$$

Then integrate both sides:

$$\int \frac{1}{y}\,dy=\int x\,dx$$

which gives:

$$\ln |y|=\frac{x^2}{2}+C$$

So:

$$y=Ae^{x^2/2}$$

where $A$ is a constant.

This method is important because it shows how differentiation and integration work together. Differentiation gives the rule for change, and integration helps recover the function from that rule.

Let’s look at a simple interpretation. If $\frac{dy}{dx}=xy$, then when $x$ is positive and $y$ is positive, the function increases. The increase gets faster as either $x$ or $y$ gets larger. This is a mathematical way to describe accelerating growth.

Interpreting constants and initial conditions

A model is not finished until its constants are interpreted. In a differential equation, constants often represent measurable quantities like growth rate, decay rate, or environmental limits. students, if a model contains a parameter $k$, you should be able to explain what happens when $k$ changes.

For example, in $\frac{dQ}{dt}=-kQ$, a larger $k$ means faster decay. In a radioactive decay model, this means the substance decreases more quickly. In finance, a larger continuous interest rate means faster growth.

Initial conditions are also essential. An equation like:

$$\frac{dQ}{dt}=-kQ$$

has many solutions, but if you know $Q(0)=Q_0$, then the solution becomes:

$$Q(t)=Q_0e^{-kt}$$

The initial condition determines the specific curve that matches the real situation.

This is one reason differential equations are such useful models: they can describe a whole family of behaviours, and then data can choose the one that fits.

Applications and connections to calculus

Modelling with differential equations connects strongly to other parts of calculus. Derivatives describe rates of change, which is the starting point for the model. Integration helps solve the equations and sometimes compute accumulated change. Graphs help you see whether a model is increasing, decreasing, or approaching an equilibrium.

This topic also connects to kinematics. If position is $s(t)$, then velocity is $v(t)=\frac{ds}{dt}$ and acceleration is $a(t)=\frac{dv}{dt}=\frac{d^2s}{dt^2}$. A model of motion may be written as a differential equation for velocity or position. For example, in basic drag models, acceleration may depend on velocity, giving equations that describe how an object slows down.

Differential equations also support optimisation and numerical reasoning. Some real models cannot be solved neatly, so you may need to use technology or approximation methods. Even when an exact solution is available, understanding the shape of the graph and the meaning of each parameter is crucial.

In IB AA HL, you should be ready to explain what a differential equation says in words, set up the equation from a context, solve simple cases, and interpret the result in the original situation.

Conclusion: Why this matters in HL calculus

Modelling with differential equations shows the real power of calculus: it turns change into prediction. Instead of only studying derivatives and integrals as separate techniques, you use them together to describe and understand real systems. Whether you are modelling population growth, cooling, decay, or motion, the same core idea appears: the rate of change determines the behaviour of the whole system.

For students, the main takeaway is this: if you can read a changing situation, identify the variable, write a rule for the rate of change, and use an initial condition, you can build a mathematical model. That skill is central to IB Mathematics: Analysis and Approaches HL and connects directly to the wider study of calculus.

Study Notes

A differential equation is an equation involving a function and one or more of its derivatives.
Ordinary differential equations use derivatives with respect to one variable, such as $\frac{dy}{dt}$.
A model often begins by stating how the rate of change depends on the current quantity.
Exponential growth and decay are modelled by equations like $\frac{dy}{dt}=ky$ and $\frac{dQ}{dt}=-kQ$.
Initial conditions such as $y(0)=y_0$ determine a unique solution.
Separable equations can be rearranged so variables are on different sides before integrating.
Equilibria occur when the rate of change is zero.
Newton’s law of cooling uses $\frac{dT}{dt}=-k(T-T_{\text{room}})$.
The logistic model $\frac{dP}{dt}=rP\left(1-\frac{P}{K}\right)$ describes growth with a carrying capacity.
Differential equations connect differentiation, integration, graphs, and real-world interpretation.
In IB AA HL, you should be able to set up, solve, and interpret simple differential equation models.