Mathematical Tools for Physics

4—Differential Equations 99

4.6 Separation of Variables
If you have a first order differential equation — I’ll be more specific for an example, in terms ofxandt— and if
you are able to move the variables around until everything involvingxanddxis on one side of the equation and
everything involvingtanddtis on the other side, then you have “separated variables.” Now all you have to do
is integrate.
For example, the total energy in the undamped harmonic oscillator isE=mv^2 /2+kx^2 / 2. Solve fordx/dt
and
dx
dt

=

√

2

m

(

E−kx^2 / 2

)

To separate variables, multiply bydtand divide by the right-hand side.

dx

√

2

m

(

E−kx^2 / 2

)=dt

Now it’s just manipulation to put this into a convenient form to integrate.

√

m

k

dx

√

(2E/k)−x^2

=dt, or

∫

dx

√

(2E/k)−x^2

=

∫ √

k

m

dt

Make the substitutionx=asinθand you see that ifa^2 = 2E/kthen the integral on the left simplifies.

∫

acosθ dθ

a

√

1 −sin^2 θ

=

∫ √

k

m

dt so θ= sin−^1

x

a

=ω 0 t+C

or x(t) =asin(ω 0 t+C) where ω 0 =

√

k/m

An electric circuit with an inductor, a resistor, and a battery has a differential equation for the current flow:

L

dI

dt

+IR=V 0 (24)