Inverse laplace transforms

Inverse laplace transforms

Lessons

To solve a differential equation using a Laplace Transform it will also be necessary to know how to use an Inverse Laplace Transform

Finding the Inverse Laplace Transform involves turning the Laplace Transform of a function back into that function.

L1F(s)L^{-1}{F(s)} = f(t)f(t)

For example from the section Calculating Laplace Transforms we saw,

LL{3e3te^{3t}} = 3s3\frac{3}{s - 3}

So to find out the inverse of the Laplace Transform of 3s3\frac{3}{s - 3}:

L1L^{-1}{3s3\frac{3}{s-3}} = 3e3te^{3t}

The same rule of Linearity applies:

L1L^{-1}{aF(s)a F(s) + bG(s)bG(s)} = aL1aL^{-1}{F(s)F(s)} + bL1bL^{-1} {G(s)G(s)}

  • 1.
    What are inverse Laplace Transforms?

  • 2.
    Inverse Laplace Transforms

    Find the inverse Laplace Transform of the following functions:

    a)
    F(s)F(s) = 12s4\frac{12}{s^{4}} - 4s6\frac{4}{s - 6} + 3s\frac{3}{s}

    b)
    G(s)G(s) = 43s212\frac{4}{3s^{2} - 12} - 3(s5)3\frac{3}{(s - 5)^{3}}


  • 3.
    Calculate the inverse Laplace Transform of the following functions:
    a)
    G(s)G(s) = 7s3s22\frac{7s - 3}{s^{2} - 2}

    b)
    H(s)H(s) = 3s5s24s+7\frac{3s - 5}{s^{2} - 4s + 7}


  • 4.
    For each of the following functions calculate their inverse Laplace Transform:
    a)
    F(s)F(s) = 7s+3(s3)(s+7)\frac{7s + 3}{(s - 3)(s + 7)}

    b)
    G(s)G(s) = 3s2+4s7(s2+7)(s3)\frac{3 s^{2} + 4s - 7}{(s^{2} + 7)(s - 3)}