Differential Equations
The subject of ordinary differential equations encompasses such a large field that you can make a profession of
it. There are however a small number of techniques in the subject that youhave to know. These are the ones
that come up so often in physical systems that you need not only the skills to use them but the intuition about
what they will do. That small group of methods is what I’ll concentrate on in this chapter.
4.1 Linear Constant-Coefficient
A differential equation such as
(
d^2 x
dt^2
) 3
+t^2 x^4 + 1 = 0is not one that I’m especially eager to solve, and one of the things that makes it difficult is that it is non-linear.
This means that if I have two solutionsx 1 (t)andx 2 (t)then the sumx 1 +x 2 is not a solution; look at all the
cross-terms you get if you try to plug the sum in to the equation. Also if you multiplyx 1 (t)by 2 you no longer
have a solution.
An equation such as
etd^3 x
dt^3+t^2dx
dt−x= 0may be a mess to solve, but if you have two solutions,x 1 (t)andx 2 (t)then the sumαx 1 +βx 2 is also a solution.
Proof? Plug in. This is called a linear, homogeneous equation because of this property. A similar-looking
equation,
etd^3 x
dt^3+t^2dx
dt−x=tdoes not have this property, though it’s close. It’s called a linear, inhomogeneous equation. Ifx 1 (t)andx 2 (t)are
solutions to this, then if I try their sum as a solution I get 2 t=t, and that’s no solution, but it misses working
only because of the single term on the right.
One of the most common sorts of differential equations that you see is an especially simple one to solve.
That’s part of the reason it’s so common. This is the linear, constant-coefficient, differential equation. If you
have a mass tied to the end of a spring and the other end of the spring is fixed, the force applied to the mass by
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