Up to now if I gave you an equation, and asked you to solve it for x you would be, in general, looking for a value of x which solved the equation. Given:

x^2+3x+2=0

You can solve this equation to find two values of x.

I could also give you an equation which linked x and y explicitly, and you could find a relationship between the two which, given a value of x would give you a value of y. You’ve been doing this now for many years. Now we’re going to add a hugely powerful tool to our mathematical arsenal. We’re going to allow our equations to include information about gradients of the function…let’s see what this means…

We’re going to take everything that you learnt about integration and turn it into a way to model and understand the world around us. This is a very powerful statement and indeed differential equations are without a doubt the most powerful mathematical tool we have to understand the behaviour of everything from fundamental particles to populations, economies, weather, flow of wealth, heat, fluids, the motion of planets, the life of stars, the flight of an aircraft, the trajectory of a meteor, the way a pendulum swings, the way a ponytail swings (see paper on this here), the way fish move, the way algae grow, the way a neuron fires, the way a fire spreads…and so much more.

So what is a differential equation? It is an equation which contains one or more derivatives of a function.

Let’s look at a very simple example, of population growth. We might want to

x^2+3x+2=0

You can solve this equation to find two values of x.

I could also give you an equation which linked x and y explicitly, and you could find a relationship between the two which, given a value of x would give you a value of y. You’ve been doing this now for many years. Now we’re going to add a hugely powerful tool to our mathematical arsenal. We’re going to allow our equations to include information about gradients of the function…let’s see what this means…

We’re going to take everything that you learnt about integration and turn it into a way to model and understand the world around us. This is a very powerful statement and indeed differential equations are without a doubt the most powerful mathematical tool we have to understand the behaviour of everything from fundamental particles to populations, economies, weather, flow of wealth, heat, fluids, the motion of planets, the life of stars, the flight of an aircraft, the trajectory of a meteor, the way a pendulum swings, the way a ponytail swings (see paper on this here), the way fish move, the way algae grow, the way a neuron fires, the way a fire spreads…and so much more.

So what is a differential equation? It is an equation which contains one or more derivatives of a function.

Let’s look at a very simple example, of population growth. We might want to

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