The Damped Harmonic Oscillator

Consider the differential equation

\frac{d^{2}y}{dt^{2}} + 2ϵ \frac{dy}{dt} + y = 0.

For definiteness, consider the initial conditions

y(0) = 0, y′(0) = 1.

Try

y = y_{0}(t) + ϵy_{1}(t) + ϵ^{2}y_{2}(t) + · · · .

We get (to second order)

\begin{aligned}0 & ≈ (y′′_{0} + ϵy′′_{1} + ϵ^{2}y′′_{2}) + (2ϵy′_{0} + 2ϵ^{2}y′_{1}) + (y_{0} + ϵy_{1} + ϵ^{2}y_{2}) \\ & = (y′′_{0} + y_{0}) + ϵ(y′′_{1} + 2y′_{0} + y_{1}) + ϵ^{2}(y′′_{2} + 2y′_{1} + y_{2}). \end{aligned}

The initial conditions don’t depend on ϵ, so they break into y_{j}(0) = 0 and

y′_{0}(0) = 1,  y′_{1}(0) = 0,  y′_{2}(0) = 0.

Now we set the coefficient of each power of ϵ equal to 0 and apply the corresponding boundary condition.

The solution for y_{0} is

y_{0}(t) = sin t.

Substitute this into the equation for y_{1} :

y′′_{1} + y_{1} = −2y′_{0} = −2 cos t.

Now remember the method of undetermined coefficients for an inhomogeneous linear equation with the forcing term “on resonance”:

y_{1} = A t cos t + B t sin t + homogeneous solution.

After algebra, you find that the solution satisfying the null initial conditions is

y_{1}(t) = −t sin t.

(Check it.) This is called a secular term, because it grows with t. For the second-order term we get the equation

y′′_{2} + y_{2} = −2y′_{1} = 2 sin t + 2 t cos t.

We know that the solution will involve t^{2} times a trig function. And so on to higher orders. (The secular terms are getting worse!) So we have constructed

y(t; ϵ) = sin t − ϵt sin t + term involving ϵ^{2}t^{2} + · · · .

To judge this approximation, let’s look at the exact solution. It is

y(t; ϵ) = \frac{e^{−ϵt}}{\sqrt{1 − ϵ^{2}}} sin(\sqrt{1 − ϵ^{2}} t).

Expanding this in a Taylor series in ϵ (with t fixed), we get agreement with our perturbative solution, as far as we’ve carried it. (Work it out.) For any given t, our approximation is good if ϵ is sufficiently small. But for a fixed ϵ, there eventually comes a t for which the error is large. Our method has led us to expand e^{−ϵt} as a power series, but that is a bad thing to do, clearly. More advanced techniques of perturbation theory are needed to get around this problem.