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

  1. #1

    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!

  2. #2
    Elite Member stapel's Avatar
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    Feb 2004
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    Cool

    Quote Originally Posted by MrMaths View Post
    This exercise is about input and output signals, with the following formula given:

    . . . . .[tex]\dfrac{dy}{dt}\, +\, y\, =\, \dfrac{dw}{dt}[/tex]



    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!

  3. #3
    Quote Originally Posted by stapel View Post
    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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