First-Order Differential Equations

A differential equation is an equation involving an unknown function and its derivatives. Instead of asking "which number?", it asks "which function?". For example,

$$y' = 2y$$

asks for every function whose derivative is twice itself. (The answer: $y = Ce^{2x}$ for any constant $C$.)

When the unknown is a function of one variable, it's an ordinary differential equation (ODE). Equations with partial derivatives are partial differential equations (PDEs), the subject of Chapter 18.

Order

The order is the highest derivative that appears.

Equation	Order
$y' = x^2 - y$	1
$y'' + 4y = 0$	2
$y''' + xy' = e^x$	3

Checking a solution

To check that a function solves an equation, substitute it. Is $y = e^{2x}$ a solution of $y' = 2y$? Its derivative is $2e^{2x}$, which is $2y$. Yes.

Is $y = \sin(3x)$ a solution of $y'' + 9y = 0$? $y'' = -9\sin(3x)$, so $y'' + 9y = 0$. Yes.

General and particular solutions

A first-order equation usually has a whole family of solutions with one arbitrary constant: the general solution. Here are a few members of the family $y = Ce^{x}$ for $y' = y$, drawn on its slope field (a short segment of slope $y'$ at each point; more in 17.05):

An initial condition such as $y(0) = 3$ picks out one curve: $3 = Ce^0$, so $C = 3$ and $y = 3e^x$. The equation plus the condition is an initial value problem (IVP), and its answer is a particular solution. An $n$th-order equation needs $n$ conditions, typically $y(x_0), y'(x_0), \dots, y^{(n-1)}(x_0)$.

Linear equations

An equation is linear if $y$ and its derivatives appear only to the first power, multiplied by functions of $x$:

$$a_n(x)y^{(n)} + \dots + a_1(x)y' + a_0(x)y = g(x)$$

It's homogeneous if $g(x) = 0$. So $y'' + x^2y = \sin x$ is linear; $y' = y^2$ and $y'' + \sin y = 0$ are not. Linear equations have a complete theory (17.03, 17.06–17.08); nonlinear ones often need numerical methods (17.05).

Where they come from

Laws of nature usually say how something changes, which is a differential equation.

Model	Equation
growth or decay at a rate proportional to size	$y' = ky$
Newton's law of cooling (room temperature $T_r$)	$T' = -k(T - T_r)$
falling with air resistance	$v' = g - kv$
logistic growth up to a capacity $K$	$P' = rP\left(1 - \frac{P}{K}\right)$
a mass on a spring	$mx'' + kx = 0$

A constant solution is an equilibrium: for cooling, $T = T_r$ (the coffee that's already at room temperature stays there).

One solution, and only one

Existence and uniqueness. If $f$ and $\partial f/\partial y$ are continuous near $(x_0, y_0)$, the problem $y' = f(x, y)$, $y(x_0) = y_0$ has exactly one solution near $x_0$. So solution curves never cross, which is why the curves above stack neatly.