IB Maths AI 5.8 Notes

Differential equations

A differential equation is an equation involving a function and one of its derivatives. Instead of giving the quantity directly, it tells us how that quantity changes. This makes differential equations very useful for modelling growth, decay, motion, populations and many other changing systems.

A first-order differential equation involves only the first derivative, usually written as

$\frac{dy}{dx}$

A general first-order differential equation can be written as

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

For example, if $y$ represents a population at time $t$, then $\frac{dy}{dt}$ represents the rate at which the population is changing. A differential equation links the size of the population to its rate of change.

However, forming a differential equation from a problem is difficult. An easy rule is that many differential equations come from phrases such as is proportional to.

• If a quantity $y$ is proportional to itself, then $\frac{dy}{dt}=ky$, where $k$ is a constant of proportionality.
• If a quantity $G$ grows at a rate proportional to $\sqrt{G}$, then $\frac{dG}{dt}=k\sqrt{G}$, where $k$ is a constant.
• If a quantity decreases at a rate proportional to its size, then $\frac{dy}{dt}=-ky$.

The sign of the constant, or the sign written in the model, tells us whether the quantity is growing or decreasing.

Form an equation for the growth of an algae population $G$ at time $t$ is proportional to $\sqrt{G}$:

This means $\frac{dG}{dt}=k\sqrt{G}$ where $k$ is a constant.

The key idea is that a differential equation does not directly tell us the value of the function. Instead, it tells us how the function is changing. Solving the equation converts this rate information into an expression for the function itself, or at least an approximation to it.

This is why differential equations are so powerful in modelling. We often know how something changes before we know its exact formula.