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Differential Equation with Summation 
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#1
Nov612, 07:30 PM

P: 16

1. The problem statement, all variables and given/known data
[tex]y''+0.1y'+y=1+2\sum_{k=1}^{n}(1)^{k}u_{k\pi}(t)[/tex] and quiescent initial conditions. 2. Relevant equations None. 3. The attempt at a solution [tex](s^{2}+0.1s+1)Y(s)=\mathcal{L}\{1\}+2\sum_{k=1}^{n}(1)^{k}\mathcal{L}\big\{ u_{k\pi}(t)\big\}[/tex] I'm not sure if this step was correct, and how to proceed since the result of that is quite nasty. Any help would be appreciated, thanks. 


#2
Nov612, 11:18 PM

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#3
Nov612, 11:41 PM

P: 16

I'm sorry, I thought it was common notation that [itex]u_{k\pi}(t)[/itex] is the indicator function denoting the unit step function:
[tex]u_{k\pi}(t)=u_{k}(t)u_{\pi}(t)=\begin{cases} 0, & t<k\quad\text{or}\quad t\ge\pi\\ 1, & k\le t<\pi \end{cases} [/tex] As for what I got for the Laplace transform of the right, it is: [tex] \frac{1}{s}+2\sum_{k=1}^{n}(1)^{k}\cdot\frac{e^{ks}e^{\pi s}}{s} [/tex] 


#4
Nov612, 11:50 PM

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Differential Equation with Summation



#5
Nov612, 11:55 PM

P: 16

Courtesy of Brannan,
The only other possibility is that [itex]c=k\pi[/itex]. Edit: Yes I think thats actually the problem faced here.. I believe it works if I let [tex]u_{k\pi}(t)=u(tk\pi)[/tex] 


#6
Nov712, 11:02 AM

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PF Gold
P: 7,632

y''+0.1y'+y=1+2\sum_{k=1}^{n}(1)^{k}u(tk\pi)$$Is that correct? And when you say "it works", do you mean you see how to finish the question? 


#7
Nov712, 02:54 PM

P: 16

Yes, that subscript notation is definitely horrible as it caused me a lot of grief haha.
And yes, by "it works" I meant I solved the DE resulting in the solution of, [tex] y(t)=h(t)+2\sum_{k=1}^{n}(1)^{k}h(tk\pi) [/tex] where [tex] h(t)=1e^{0.05t}\cos(\sqrt{0.9975}t)\frac{0.05e^{0.05t}}{\sqrt{0.9975}}\sin(\sqrt{0.9975}t) [/tex] Thanks for the 'indirect' help! 


#8
Nov712, 03:06 PM

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[Edit]Never mind about the n. I see where it is now. 


#9
Nov712, 03:10 PM

P: 16

The next part of the question was actually to graph the forcing function and solution on the same set of coordinates. So you are interpreting it correctly.



#10
Nov712, 03:13 PM

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P: 7,632




#11
Nov712, 03:14 PM

P: 16

Yeah, we too let Maple do the grunt work. Funny thing is Maple was developed at my university so they push us to use it.



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