Find the inverse Laplace transform of 1/x^(3/2).

Published on October 27, 2024

Calculus

The inverse Laplace transform of 1/x^(3/2) is 2/(sqrt(pi))*sqrt(x).

Question

inverse\:laplace\:\frac{1}{x^{\frac{3}{2}}}

Answer

Let’s find the inverse Laplace transform of

1. Recall the definition:

The inverse Laplace transform, denoted by

, is the operation that takes a function

(in the Laplace domain) and returns a function

(in the time domain) such that:

2. Utilize the Laplace Transform table:

We need to find an entry in the Laplace Transform table that resembles our function

. Notice that the Laplace transform of

is given by:

This closely matches our given function.

3. Apply linearity and adjust for the variable:

The Laplace transform is linear. Since we are using ‘

’ instead of ‘

’, we can directly apply the inverse transform.

Therefore, the inverse Laplace transform of is:

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