Diff-EQ: integrating factors - signals (how could f(t)δ′(t) be simplified?)

MrMaths

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Diff-EQ: integrating factors - signals (how could f(t)δ′(t) be simplified?)

Hey!

I'v been stuck now for a while and found this forum - I hope you guys could help.

The question is: how could f(t)δ(t) be simplified? And this is about input and output signals.

The given formula from the question:
dy/dt + y = dw/dt




They gave me two hints:
1) Use integrating factors
2) There is a integration constant which you could determine since the system is causal

My solution (that I believe is wrong):

I believe the integrating factor is e^t since we have 1*y.

Step 1: d/dx*(y*e^t) = dw/dt * e^t
Step 2: y*e^t = integral | dw/dt * e^t dt = e^t + c
Step 3: y * e^t = e^t + c
Step 4: y = 1 + c*e^-1

The confusing part is step 2. The integral of dw/dt. I believe this is my issue right there.

Also I fail to see how I could reach f(t)δ(t) from here. I know y = Sw.

Any help would do. Thanks!
 
This exercise is about input and output signals, with the following formula given:

. . . . .\(\displaystyle \dfrac{dy}{dt}\, +\, y\, =\, \dfrac{dw}{dt}\)



How could f(t) δ′(t) be simplified?

Hints:
1) Use integrating factors.
2) There is a integration constant which you could determine since the system is causal.




My solution (that I believe is wrong):

I believe the integrating factor is e t since we have 1*y.

Step 1: d/dx*(y*e^t) = dw/dt * e^t
Step 2: y*e^t = integral | dw/dt * e^t dt = e^t + c
Step 3: y * e^t = e^t + c
Step 4: y = 1 + c*e^-1

The confusing part is step 2. The integral of dw/dt. I believe this is my issue right there.

Also I fail to see how I could reach f(t) δ′(t) from here. I know y = Sw.
What are f, δ, y, t, w, x, and S? How do they relate, if at all?

Thank you! ;)
 
What are f, δ, y, t, w, x, and S? How do they relate, if at all?

Thank you! ;)

Not sure about if this was a question to help you or to help me. In any case I do not see how all of these would help me understand my two specific questions:
1) The confusing part is step 2. The integral of dw/dt. I believe this is my issue right there.

2) Also I fail to see how I could reach f(t) δ′(t) from here. I know y = Sw.

Anyone who knows?
 
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