Differential Equations: Definitions

A differential equation is simply an equation where the unknown is a whole function, and the equation tells you how that function changes. It's called "differential" because it's built from derivatives, and $dy$ and $dt$, the pieces of the derivative $\frac{dy}{dt}$, are called differentials. It's how scientists write down nearly every law about change. The general linear one in the formula box looks like a wall of symbols, but all it says is: take $y$ and some of its derivatives, up to $y^{(n)}$ (the $n$th derivative, the highest one present), multiply each by a coefficient like $a_1(t)$ (a number or a function of $t$), add them up, and set the total equal to $g(t)$ (the outside push, which has no $y$ in it). It's just a few simple parts added together, so bear with it.

You've already met a few of these without the name. Newton's second law, $m\,y'' = F$, is a differential equation: it doesn't hand you the position $y$, it tells you what its second derivative has to be. Population growth $P' = kP$ is another one: it says the population grows at a rate proportional to its current size. Most of physics, biology, economics and engineering is written this way, because it's usually much easier to say how something changes than to say what it is.

Vocabulary. There are a handful of words you'll see all course long. They're just ways of sorting equations by how hard they are and which methods work on them.

• Dependent and independent variable: the unknown function is the dependent variable, and what it's a function of is the independent variable. The letters don't matter, only the roles: in $m''' + o\,m = \tan o$, the unknown function is $m(o)$, so $m$ is dependent and $o$ is independent.
• Ordinary (one independent variable) vs partial (several, with partial derivatives).
• Order: the highest derivative. $y''' + y = 0$ is third order.
• Linear: $y$ and its derivatives appear only to the first power, not multiplied together or inside other functions. Their coefficients may depend on the independent variable. $y'' + ty = \sin t$ is linear; $y' = y^2$ and $y'' + \sin y = 0$ are not.
• Homogeneous (for a linear equation): every term contains the unknown function or one of its derivatives. A leftover term $g(t)$ with no $y$ in it, even a constant, makes it nonhomogeneous.
• Constant coefficients: the coefficients of $y, y', y'', \dots$ are all constants. $y'' + 3y' + 2y = e^t$ has constant coefficients; $y'' + ty = 0$ doesn't.
• Autonomous: the independent variable never appears on its own, as in $y' = y(1 - y)$ or $y'' + y = 5$. Any lone $t$, like in $y' = y + t$, makes it nonautonomous.

A common mistake. In $y'' + ty = \sin t$, it's tempting to think the equation is nonlinear because of the $t$ multiplying $y$ and the $\sin t$ on the right. It isn't. Linearity is only about how $y$ shows up. The $t$ is just a coefficient, and $\sin t$ has no $y$ in it at all. What breaks linearity is doing something to $y$ itself, like squaring it in $y' = y^2$ or putting it inside a sine in $y'' + \sin y = 0$.

Solutions. A function is a solution if substituting it makes the equation true. Think about what that means for $P' = kP$: any $Ce^{kt}$ works, whatever $C$ is. That's typical. The general solution contains arbitrary constants (one per order). An initial value problem (IVP) adds conditions like $y(0) = 5$ that pin them down, and what you get is a particular solution.

Checking is easy. Plug the candidate in. That's how you'll verify every answer in this course.

Every simulation, from weather forecasts to epidemics to the spring in a car's suspension, starts by writing down a differential equation for how things change.