Here’s a starter.
You probably know what a graph is. Let’s say that y is the vertical axis and x the horizontal axis, and the graph is a joined up picture of a whole collection of points relating a value of y to a value of x. The graph may be represented by the equation y=f(x), where f(x) is a function of x containing terms involving x and powers of x. The graph is a wiggly line and at any point on it we should be able to draw a tangent to the point, which is a line perpendicular to the point. That tangent at a point is the gradient and can be represented by dy/dx. We should be able to actually (in theory) find the gradient for every point on the line. We could then draw another graph with the gradient values as the vertical axis instead of y. The x axis stays the same. The new graph could be written dy/dx=g(x) where g(x) is a different function of x. f(x) and g(x) are related, and g(x) is called the differential of f(x), so we could write g(x)=df/dx, and g(x)=dy/dx, both of which are examples of differential equations.
Given dy/dx=g(x) and using certain rules we should be able to work out y=f(x). But, we can’t do this with accuracy without further information, because different graphs can have the same gradient at all points.
Also, we can include y in a differential equation so we may have y+dy/dx=g(x), making the solution y=f(x) harder to find. Again, there are tricks and rules to help you solve this and find out what f(x) is.
And we can take gradients further by finding the gradient of dy/dx, giving us the next order d²y/dx². Another equation arises: d²y/dx²=h(x), another function. A second order differential equation might look like ad²y/dx²+bdy/dx+y=p(x), where a and b could be constants or functions of x themselves and p(x) is another function of x or a constant. Tricks and rules are needed to solve this to end up with y=f(x), but, again, more info is needed to find an explicit accurate determination of f(x).
Finally, what in the real world is a gradient? It’s a rate of change. An example is rate of change of position which is called velocity, so dy/dx would be a velocity. Or it could be the rate of growth of a population of people, rabbits, flies, whatever. Differential equations when solved can tell you how a population grows, for example.
The rate of change of velocity is acceleration, represented by d²y/dx² where y is position and x is time. A differential equation containing this would, for example, lead you to find the position of a body under the influence of gravity.
I hope this introduction is useful.
