We wish to solve the problem
PDE: $\quad w_t = 3^2 w_{xx} , \qquad -\infty < x < \infty , \quad t > 0$
IC: $\quad w(x,0) = 8 \sin( 6 x )$
using the Laplace transform.

First if we let $W(x,s) = W(x) = \mathcal{L}[w]$ with Laplace transforming the time variable, then the equation becomes (use only $W$ in your answer, not $W(x)$)
${} = {}$ $\frac{d^2W}{dx^2}$

The solution then is $w(x,t) = {}$

help (formulas)