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What are differential equations, and why are they important?

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’!

 

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