Green's Function for Third Order ODE

[ferret123]

Homework Statement

A dynamical system has a response, y(t), to a driving force, f(t), that satisfies a differential equation involving a third time derivative:

$\frac{d^{3}y}{dt^{3}} = f(t)$

Obtain the solution to the homogeneous equation, and use this to derive the causal Green's function for this system, G(t;τ). [hint: which order of derivative has a discontinuity at t = τ?]

2. The attempt at a solution

I've obtained a solution to the homogeneous equation $\frac{d^{3}y}{dt^{3}} = 0$ by integrating 3 times with respect to t giving $y(t) = \frac{1}{2}At^{2} + Bt + C$.

Since I'm looking for a causal Green's function I know for t<τ G(t;τ) = 0.

Taking the advise of the hint I have tried to find which order of derivative has a discontinuity at t=τ. First replacing the driving force, f(t), with a delta function, δ(t-τ), I get

$\frac{d^{3}y}{dt^{3}} = \delta(t-τ)$

then integrating over the interval [τ-ε, τ+ε] and letting ε tend to 0, I conclude the second derivative changes discontinuously by 1.

Is this correct or have I missed something in determining where the discontinuity lies?
Thanks.

 

[vela]

Looks fine.