Mathematical Tools for Physics

4—Differential Equations 85

All you have to do is to recall that the derivative of an exponential is an exponential. det/dt=et. If I
substitute this exponential forx(t), of course it can’t work as a solution; it doesn’t even make sense dimensionally.
What iseto the power of a day? You have to have something in the exponent to make it dimensionless,eαt. Also,
the functionxis supposed to give you a position, with dimensions of length. Use another constant:x(t) =Aeαt.
Plugthisinto the differential equation ( 4 ) to find

mAα^2 eαt+bAαeαt+kAeαt=Aeαt[mα^2 +bα+k] = 0

The product of factors is zero, and the only way that a product of two numbers can be zero is if one of the
numbers is zero. The exponential never vanishes, and for a non-trivial solutionA 6 = 0, so all that’s left is the
polynomial inα.

mα^2 +bα+k= 0, with solutions α=

−b±

√

b^2 − 4 km

2 m

(5)

The position function is then
x(t) =Aeα^1 t+Beα^2 t (6)

whereAandBare arbitrary constants andα 1 andα 2 are the two roots.
Isn’t this supposed to be oscillating? It is a harmonic oscillator after all, but the exponentials don’t look
very oscillatory. If you have a mass on the end of a spring and the entire system is immersed in honey, it won’t do
much oscillating! Translated into mathematics, this says that if the constantbis too large, there is no oscillation.
In the equation forα, ifbis large enough the argument of the square root is positive and bothα’s are real — no
oscillation. Only ifbis small enough does the argument of the square root become negative; then you get complex
values for theα’s and hence oscillations.
Push this to the extreme case where the damping vanishes:b= 0. Thenα 1 =i

√

k/mandα 2 =−i

√

k/m.

Denoteω 0 =

√

k/m.

x(t) =Aeiω^0 t+Be−iω^0 t (7)

You can write this in other forms, see problem 10. To determine the arbitrary constantAandByou need two
equations. They come from some additional information about the problem, typically some initial conditions.
Take a specific example in which you start from the origin with a kick,x(0) = 0andx ̇(0) =v 0.

x(0) = 0 =A+B, x ̇(0) =v 0 =iω 0 A−iω 0 B