Modelling with Differential Equations

students, imagine trying to predict how quickly a virus spreads in a school, how a tank fills with water, or how the temperature of a hot drink changes over time ☕. In each case, the key question is not just what happens, but how fast it changes. Differential equations are one of the main tools mathematicians use to describe change in the real world.

In this lesson, you will learn how differential equations connect to calculus, how they are used to build models, and how to interpret them in context. By the end, you should be able to explain the meaning of terms like $\frac{dy}{dx}$, recognize when a situation can be modelled by a differential equation, and understand what a solution means in a real setting.

What a Differential Equation Means

A differential equation is an equation that includes a derivative, such as $\frac{dy}{dx}$ or $\frac{dy}{dt}$. It describes a relationship between a quantity and the rate at which that quantity changes.

For example, if $y$ is the amount of water in a tank and $t$ is time, then $\frac{dy}{dt}$ tells us how quickly the amount of water is changing. If the water is flowing in, $\frac{dy}{dt}$ may be positive. If the tank is leaking, $\frac{dy}{dt}$ may be negative.

A differential equation does not usually give a single number straight away. Instead, it gives a rule for change. Solving the differential equation gives a function that matches that rule.

A simple example is

$$\frac{dy}{dx}=3x^2.$$

This means the rate of change of $y$ with respect to $x$ is $3x^2$. A function that fits this is $y=x^3+C$, because

$$\frac{d}{dx}(x^3+C)=3x^2.$$

Here, $C$ is a constant of integration. This shows the connection between differentiation and integration: if you know the rate of change, you can often recover the original quantity by integrating.

Why Differential Equations Are Useful in Real Life

Differential equations are useful because many real situations depend on rates of change, not just fixed values. They help model situations where one quantity influences another over time or distance.

Here are some common examples:

Population growth 🐇: the growth rate may depend on the current population.
Cooling of a drink 🧊: the temperature changes depending on the difference between the drink and the room.
Motion 🚗: velocity is the derivative of position, and acceleration is the derivative of velocity.
Medicine 💊: the amount of a drug in the body may decrease at a rate proportional to the amount present.

In IB Mathematics: Applications and Interpretation HL, the goal is not just to solve equations mechanically. You must also interpret the model, state assumptions, and judge whether the result makes sense in context.

For example, if a model predicts a population becomes negative, that is a sign the model should only be used over a certain range or that its assumptions are unrealistic.

Building a Model from a Rate Rule

A mathematical model often begins with a statement like: “the rate of change is proportional to the amount present.” This is one of the most important modelling ideas in calculus.

If $y$ is the amount of a substance, then “rate proportional to amount” can be written as

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

where $k$ is a constant.

This type of equation appears in many situations:

If $k>0$, the quantity grows exponentially.
If $k<0$, the quantity decays exponentially.

To see why, separate the variables:

$$\frac{1}{y}dy=k\,dt.$$

Integrating both sides gives

$$\ln|y|=kt+C,$$

$$y=Ae^{kt}.$$

This is the general solution, where $A=e^C$ is a constant.

If an initial condition is given, such as $y(0)=50$, then you can find $A$:

$$50=Ae^{k\cdot 0}=A,$$

so the model becomes

$$y=50e^{kt}.$$

This is a very common way differential equations are used in practice: a real-world statement becomes an equation, and then an initial condition makes the model specific.

Interpreting Solutions in Context

A solution to a differential equation only becomes meaningful when you connect it to the situation being modelled. students, this is where careful interpretation matters most.

Suppose a bacteria culture is modelled by

$$\frac{dN}{dt}=0.4N,$$

where $N$ is the number of bacteria and $t$ is time in hours.

The solution is

$$N=Ne^{0.4t},$$

but that is not yet complete because the constant must be found from an initial value. If $N(0)=200$, then

$$N(t)=200e^{0.4t}.$$

This means:

at $t=0$, there are $200$ bacteria;
the population is increasing continuously;
the rate of increase gets larger as the population gets larger.

It is important to state the units. If $t$ is in hours, then $0.4$ has units of $\text{hour}^{-1}$.

Another key skill is identifying what the model does and does not say. The equation does not explain every biological detail. It only claims that growth rate is proportional to population size. That assumption is powerful, but also simplified.

A Classic Example: Exponential Growth and Decay

Exponential models are among the most important differential equation models in IB calculus.

Growth

If a population grows according to

$$\frac{dP}{dt}=kP, \quad k>0,$$

then the solution is

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

where $P_0$ is the initial population.

This model is often used for idealized growth, such as bacteria under perfect conditions.

Decay

If a radioactive substance decays according to

$$\frac{dA}{dt}=-kA, \quad k>0,$$

then the solution is

$$A(t)=A_0e^{-kt}.$$

This shows that the amount decreases quickly at first, then more slowly over time.

A common IB-style question may ask for the half-life. If $A(t_{1/2})=\frac{A_0}{2}$, then

$$\frac{A_0}{2}=A_0e^{-kt_{1/2}}.$$

Dividing by $A_0$ gives

$$\frac{1}{2}=e^{-kt_{1/2}}.$$

Taking natural logs,

$$\ln\left(\frac{1}{2}\right)=-kt_{1/2},$$

$$t_{1/2}=\frac{\ln 2}{k}.$$

This type of reasoning shows how calculus helps link a rule for change with a meaningful quantity like half-life.

Solving and Using Differential Equations with Technology

In HL Applications and Interpretation, technology is often used to explore differential equations numerically and graphically. This is very important because many realistic differential equations cannot be solved exactly with simple algebra.

For instance, a model may be too complex for a neat formula like $y=Ae^{kt}$. In those cases, technology can help by:

drawing slope fields;
estimating solutions numerically;
comparing a model with data;
checking whether a proposed solution is reasonable.

A slope field shows short line segments representing the slope $\frac{dy}{dx}$ at many points. If the equation is

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

then the slope at each point depends on both $x$ and $y$. By following the pattern of the segments, you can sketch approximate solution curves.

Technology can also solve initial value problems numerically, such as

$$\frac{dy}{dt}=ky(1-y),$$

which is a logistic model. Unlike exponential growth, logistic growth includes a limiting value, often called a carrying capacity. This is useful for populations that cannot grow forever because of limited resources.

When using technology, you should always interpret the output carefully. A graph or table may show a trend, but the model still depends on assumptions and parameters.

Modelling Steps You Should Remember

When working with differential equations in IB Mathematics: Applications and Interpretation HL, a structured approach is helpful.

Identify the changing quantity and the independent variable.
Translate the real situation into a statement about rate of change.
Write the differential equation.
Use an initial condition or boundary condition if one is given.
Solve the equation, exactly or numerically.
Interpret the result in context.
Check whether the answer is sensible.

For example, if a hot drink cools in a room, a common model is Newton’s law of cooling:

$$\frac{dT}{dt}=-k(T-T_s),$$

where $T$ is the drink temperature, $T_s$ is the surrounding temperature, and $k>0$.

This says the rate of cooling depends on the difference between the object and its surroundings. If the drink is much hotter than the room, it cools faster. As it gets closer to room temperature, the cooling slows down.

That is a perfect example of how a differential equation turns a real pattern into a mathematical rule.

Conclusion

Differential equations are a central part of calculus because they connect derivatives, rates of change, and real-world accumulation. students, they help answer questions like how fast something changes, how a system evolves, and what pattern best fits observed data. In IB Mathematics: Applications and Interpretation HL, you are expected to understand both the mathematics and the context.

The most important ideas are: a differential equation describes a rate rule, a solution gives a function that fits the rule, and an initial condition makes the model specific. Technology is valuable for exploring models that are difficult to solve by hand, but interpretation remains essential. Good modelling means combining mathematical methods with clear reasoning about the real situation.

Study Notes

A differential equation is an equation involving a derivative, such as $\frac{dy}{dx}$ or $\frac{dy}{dt}$.
Differential equations describe how a quantity changes, not just its value.
A solution is a function that satisfies the differential equation.
Initial conditions help determine the constant in the general solution.
A model is often built from a real statement like “the rate is proportional to the amount.”
If $\frac{dy}{dt}=ky$, then $y=Ae^{kt}$.
If $\frac{dy}{dt}=-ky$, then $y=Ae^{-kt}$.
Exponential growth happens when $k>0$; exponential decay happens when $k<0$.
Units matter: if $t$ is measured in hours, then parameters involving $t$ must match those units.
Interpret solutions in context and check whether the result is realistic.
Technology can help with slope fields, numerical solutions, and graphing.
Modelling with differential equations links differentiation, integration, and real-world change in one topic.