# 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:

*e*^{kx}* = y*

Thus we have shown that the exponential function *y(x) = e*^{kx} 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)** ∂^{2}ψ/∂x^{2} = (1/v^{2})(∂^{2}ψ/∂t^{2})

**(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 - **ω**t**)*

Let us verify this.

* Calculate **(4)**’s second partial derivative with respect to *x* (we will need the Chain Rule):

**(5)** ∂^{2}ψ/∂x^{2}*= -k ^{2}*

*Asin(kx -*

*ω*

*t*

*)*

* Calculate **(4)**’s second partial derivative with respect to *t*:

**(6)** ∂^{2}ψ/∂t^{2}*= -ω*^{2}*Asin(kx - **ω**t**)*

* 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:

*-k*^{2}*Asin(kx - **ω**t**) = -**(1/v*^{2}*)**ω*^{2}*Asin(kx - **ω**t**)*

*-k*^{2}* = -(1/v*^{2}*)**ω*^{2}

*k*^{2}*/**ω*^{2}* = 1/v*^{2}

*v*^{2}* = **ω*^{2}*/k*^{2}

*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 - **ω**t**)* 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’!