Mathematical Tools for Physics

6—Vector Spaces 150

Suppose you have the relation between two functions of time

A−Bω+γt=βt

that is, that the twofunctionsare the same, think of this in terms of vectors: On the vector space of polynomials
inta basis is
~e 0 = 1, ~e 1 =t, ~e 2 =t^2 , etc.

Translate the preceding equation into this notation.

(A−Bω)~e 0 +γ~e 1 =β~e 1

For this to be valid the corresponding components must match:

A−Bω= 0, and γ=β

Differential Equations
When you encounter differential equations such as

m

d^2 x

dt^2

+b

dx

dt

+kx= 0, or γ

d^3 x

dt^3

+kt^2

dx

dt

+αe−βtx= 0, (3)

the sets of solutions to these equations form vector spaces. All you have to do is to check the axioms, and because
of the theorem in section6.3you don’t even have to do all of that. The solutions are functions, and as such they
are elements of the vector space of example 2. All you need to do now is to verify that the sum of two solutions
is a solution and that a constant times a solution is a solution. That’s what the phrase “linear, homogeneous”
means.
Another common differential equation is

d^2 θ

dt^2

+

g

`

sinθ= 0

This describes the motion of an undamped pendulum, and the set of its solutions donot form a vector space.
The sum of two solutions is not a solution.