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by WarHorseLeBron on May 12th, 2011

WarHorseLeBron

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Help answer this question below.

Find the Laplace transform of the equation with the given boundary conditions.

f''(t)+f(t) = sin(ßt)
f'(0) = 0
f(0) = 0

This is question 14/15 from my FE review manual.

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Answers. 1 helpful answer below.

  • by douglasmkilmer on May 12th, 2011
    voted: F(s) = [1/(1+s²)][ß/(s²+ß²)]

    douglasmkilmer

    Asker's Pick

    Selected by the asker, WarHorseLeBron. (What's this?)

    Given:

    f''(t)+f(t) = sin(ßt)
    f'(0) = 0
    f(0) = 0

    We will be transforminng both sides but let's consider, the left hand side first:
    f''(t) becomes s²F(s) - sf'(0) - f(0)

    but f'(0) and f(0) are both 0 so the f''(t) becomes s²F(s) and f(t) becomes F(s).

    The laplace transform of sin(ßt) can be found in most Laplace transform tables as ß/(s² + ß²)

    Therefore the equation becomes:

    s²F(s)+F(s) = ß/(s² + ß²)

    Removing the common factor from the left hand side:

    (s² + 1)F(s) = ß/(s² + ß²)

    Dividing both sides by s² + 1:

    F(s) = [1/(s² + 1)][ß/(s² + ß²)]

    This looks like the first selection.

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