Is the Solution sin(t)H(t) for SHO a Particular Solution?

[Coffee_]

So we have derived that for the differential equation:

##x(t)''+x(t)=\delta(t)##

The solution is given by ##x=sin(t)H(t)## where ##H## is the Heaviside function.

To find this we assumed that the system was in rest before ##t=0## and that position and velocity are continious.

QUESTION: I am pretty sure that this ##sin(t)H(t)## is just a particular solution, is it correct to say that if one doesn't assume that the system in rest the general solution is given by ##x=Acos(t)+Bsin(t)+H(t)sin(t)##? So basically, is the solution we found in class a particular solution and thus can I always add a homogeneous solution to it?

 

[Orodruin]

Yes, but generally you will want your Green's functions to be causal, i.e., the effect of the impulse only shows after the impulse. This is also required for using the Green's function to adapt to homogeneous initial conditions (you can also use it for inhomogeneous initial conditions using a Green's function fulfilling homogeneous initial conditions, but that is another story).

 

[Coffee_]

The reason I was asking is that for a problem of type ##q''(t)+q(t)=f(t)## we wrote down the general solution as :

##q(t)=Acos(t)+Bsin(t) + \int_{-\infty}^{+\infty} G(t-t') f(t') dt'## (1)

If the green's function was not strictly a particular solution but a general solution to ##q''(t)+q(t)=\delta(t)## then the ##Acos(t)+Bsin(t)## term wouldn't be required because it would implictly be sitting in the integral.

 

[Orodruin]

Well, since you would typically define your Green's function to be zero before the impulse, it would not be a matter of having the possibility to add any homogeneous solution (this would mess with the initial conditions required from the Green's function). If you have enough initial conditions, you cannot add things in that manner and the Green's function typically comes with enough initial conditions to determine it uniquely. You cannot add a homogeneous solution to it because this would violate the initial conditions imposed on the Green's function.

 

[Coffee_]

And hence, Green's function is only a particular solution to the ##q''(t)+q(t)=\delta{t}## problem, and thus in equation (1) above the integral is only a linear combination of particular solutions and so, I have to explictly add the homogenous part to make it most general. I think I get it thanks.

 

[Orodruin]

Yes, also note that your time interval would usually not be from minus infinity, but you would impose some initial conditions at a fixed time. These can also be taked care of with the Green's function, using it to describe boundary terms.