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Let's begin by reviewing a fairly simple (linear) equation:
(1) y = 2x + 3
(1) describes the relationship between two variables, x and y. Given a value for one, we can use (1) to work out the value of the unknown. For instance, when x=4, y=11.
Differential equations are more general. They include derivative terms, denoted dy/dx or y'. Let us consider an example:
(2) y' = dy/dx = ky
While (1) described the relation between two variables, (2) describes the relation between y's slope (with respect to x) and y itself. In English, it says that "y's slope is directly proportional to y." Unlike (1), the solution to (2) is a function y, that satisfies the above description. But what kind of function has these properties? Let us solve (2) to find out:
* Take all the 'x' terms to one side, and all the ‘y’ terms to the other side:
dy/dx = ky
dy = ky dx
dy/y = k dx
(1/y) dy = k dx
* Take the integral of both sides (we omit some details about definite integrals and boundary conditions, for simplicity):
INTEGRAL( (1/y) dy) = INTEGRAL(k dx)
* Evaluate the above integral:
ln(y) = kx
* Which is true if and only if:
ekx = y
Thus we have shown that the exponential function y(x) = ekx satisfies the differential equation (2). This is what we should expect, since (2) said "y's slope is directly proportional to y" and this is exactly how the number e is defined.
In further study, you will find that differential equations like (2) have applications in diverse fields such as: in Economics, to model growth rates and continuously compounded interest; Physics, to model radioactive decay or damped oscillations; in Biology to model bacteria populations. Now let us consider a more difficult example:
(3) ∂2ψ/∂x2 = (1/v2)(∂2ψ/∂t2)
(3) looks a little daunting but it is not much more complicated than (2). The solution to (3) needs to be a function ψ(x, t). x stands for a spatial direction, and t is time. v stands for ‘speed.’ We are given details about how ψ’s second (partial) derivatives, with respect to x and t, relate to each other. (3) is known to Physicists as the 'Wave Equation', because it can be derived by carefully studying springs in oscillation and sine/cosine waves like (4) are solutions to it.
(4) ψ = Asin(kx - ωt)
Let us verify this.
* Calculate (4)’s second partial derivative with respect to x (we will need the Chain Rule):
(5) ∂2ψ/∂x2= -k2Asin(kx - ωt)
* Calculate (4)’s second partial derivative with respect to t:
(6) ∂2ψ/∂t2= -ω2Asin(kx - ωt)
* We also need the following result (known as a dispersion relation). In physics, It can be shown that:
(7) ω = kv
where ω is ‘angular frequency’, k is ‘wavenumber’ and v is the speed that a wave propagates at.
* Substituting (5) and (6) into (3) gives:
-k2Asin(kx - ωt) = -(1/v2)ω2Asin(kx - ωt)
-k2 = -(1/v2)ω2
k2/ω2 = 1/v2
v2 = ω2/k2
kv = ω
* Which we know to be correct because of result (7), which follows from the definitions of ‘angular frequency’, ‘wavenumber’ and ‘wave speed’.
Hence the function ψ(x, t) = Asin(kx - ωt) is a solution to the differential equation (3). In fact, Schrödinger’s equation is a famous wave equation, incorporating complex numbers, whose solution is the ‘wavefunction’ of a system of particles. So I hope you can see, from this brief introduction, that we can get a lot of mileage out of this concept of ‘differential equations’!