Order and linearity · Differential Equations

Intuition

Before trying to solve an equation it is worth asking two questions about it, because between them they decide which methods can be used at all. How many times is the unknown differentiated? And does the unknown enter in a straight-line way, or is it squared, put inside a sine, multiplied by its own derivative? The first question gives the order, the second gives linearity, and almost every technique in this course is a technique for linear equations.

Order is how far back the equation remembers: a first-order equation knows only where you are, a second-order one knows where you are and how fast you are going. Linearity is a promise about the shape of the rule — that doubling the input doubles its contribution — and it is the promise that lets answers be added together.

Linear equations are a small part of all of them, and nearly every method in this course lives inside the circle. The pendulum $y''+\sin y=0$ sits outside it, which is why the ninth chapter has to look at it differently.

Order, and what linear means

The order of an equation is the highest derivative that appears in it. The equation is linear if it can be written with the unknown and its derivatives each appearing alone, to the first power, multiplied only by functions of the independent variable. Everything else is nonlinear.

a_{n}(x)y^{(n)}+\dots+a_{1}(x)y'+a_{0}(x)y=g(x)

Reading an equation

The coefficients $a_{k}$ and the right-hand side $g$ may be as complicated as they like — $x^{7}$ , $\sin$ , — and the equation is still linear. Linearity is a condition on , never on .

Applications

Practice

Order Is the Highest Derivative

Count the derivatives and take the largest. Powers change nothing about the order.

(y')^{5}+y''=x\quad\text{is second order}

Try it

What is the order of $(y')^{4}+y=x^{3}$ ?