Diff Eq and the Dirac Delta(impulse) function.

[faust9]

Ok, I was given: Solve the following using superposition:

$$\ddot{x}+2\dot{x}+4x=\delta(t)$$

bounded by $$\dot{x}=0, x(0)=0$$

I solved the Homo eqn and got the following:

$$x(t)=e^{-t}(\cos (\sqrt{3}t)+\frac{\sqrt{3}}{3}\sin ({\sqrt{3}t))$$

I also know that :
$$\ddot{x}+2\dot{x}+4x=u(t)$$

equals

$$x(t)=e^{-t}(\cos (\sqrt{3}t)+\frac{\sqrt{3}}{3}\sin ({\sqrt{3}t))+\frac{1}{4}$$ from a previous problem.

So, I said:

$$x(t)=e^{-t}(\cos (\sqrt{3}t)+\frac{\sqrt{3}}{3}\sin ({\sqrt{3}t))+\frac{\delta (t)}{4}$$

Is this correct thus far (the superposition part at least I'm pretty sure the diff eq portion is correct).

How do I deal with the delta function? Any help would be much appreciated!

 

[dextercioby]

Yes,u need to find the Green function...Do u know how to use the theorem of residues...??

Daniel.

 

[faust9]

Yes,u need to find the Green function...Do u know how to use the theorem of residues...??

Daniel.

Nope. My professor flew through this the other day (a similar problem) and looking at my notes it seems I missed something while he was talking.

edit:

Should I use the Homo eqn and the characteristic eqn for a delta function and solve using variation of parameters?

edit:edit:
I can't use the above because I only have two initial eqn's correct?

 

[dextercioby]

I don't know how to do it otherwise...It's a LINEAR second order nonhomogenous ODE and you're asked for the Green function of the LINEAR DIFFERENTIAL OPERATOR
$$\hat{O}=:\frac{d^{2}}{dt^{2}}+2\frac{d}{dt}+4$$

Maybe someone else could come up with a different solution which wouldn't require distributions and residues theorem.

Daniel.

 

[faust9]

Anyone else out there in physics land have a hint for me?

 

[Hurkyl]

Hrm, I can see two other approaches that might work.

(1) Build the solution in patches. Note that if you restrict the domain to t >= 0, or t <= 0, the differential equation is purely homogenous. Then, the goal is to pick solutions on each of these domains that, when patched together, satisfy the differential equation at 0.

(2) Write &delta;(t) as a limit of something, maybe as a limit of step functions. Then, with some hand-waving and some optimism, you can take the limit of the corresponding solutions to get a solution to your desired equation.

 

[faust9]

Hrm, I can see two other approaches that might work.

(1) Build the solution in patches. Note that if you restrict the domain to t >= 0, or t <= 0, the differential equation is purely homogenous. Then, the goal is to pick solutions on each of these domains that, when patched together, satisfy the differential equation at 0.

(2) Write ?(t) as a limit of something, maybe as a limit of step functions. Then, with some hand-waving and some optimism, you can take the limit of the corresponding solutions to get a solution to your desired equation.

Thanks! So simple... I used approach number 1 where time is defined t>=0 so there is only one instant where the delta function applies t=0.

I have:
$$x(t)=e^{-t}(\cos (\sqrt{3}t)+\frac{\sqrt{3}}{3}\sin ({\sqrt{3}t))$$

with my interval greater than t=0 as my answer.

Thanks for the help.

 

[polyb]

Anyone else out there in physics land have a hint for me?

Here are a couple of links that will give you the basics about the delta function.

http://mathworld.wolfram.com/DeltaFunction.html

http://en.wikipedia.org/wiki/Dirac_delta

Hopefully you won't get lost and should find that info relevant and useful. They cover the basic standard definitions plus some more of the delta function.

One thig I noticed off the bat was that your solution did not match your contstraint of x(0)=0. If that boundry was given, then your solution should be zero at t=0. Also it is an inhomogenous ODE, so solve it as such.

If you are so inclined and have the time, here is a link to from mathworld on Green's functions, you will be seeing them in the future.

http://mathworld.wolfram.com/GreensFunction.html

Enjoy!