# Homework Help: Diff Eq and the Dirac Delta(impulse) function.

1. Jan 26, 2005

### 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!!!

Last edited: Jan 26, 2005
2. Jan 26, 2005

### dextercioby

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

Daniel.

3. Jan 26, 2005

### faust9

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?

Last edited: Jan 26, 2005
4. Jan 26, 2005

### 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.

5. Jan 26, 2005

### faust9

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

6. Jan 26, 2005

### Hurkyl

Staff Emeritus
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.

7. Jan 26, 2005

### faust9

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.

8. Jan 26, 2005

### polyb

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 wont get lost and should find that info relevent 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!