I've been working on this problem for a while now, for context, I'm allowed to solved using Fourier integral, transform or series, but the most recent topic was the Laplace transform which is the method I've been trying.
$$\begin{cases} \dfrac{\partial w}{\partial t} = \dfrac{\partial^2 w}{\partial x^2}\ (-\infty <x<\infty ,\ t > 0)\\ u(x,0)=\exp(-|x|) \end{cases}$$
The transformed equation results in $$ sW-w(x,0)=\frac{d^2W}{dx^2} \to W''-sW=-\exp(-|x|)$$ With what I've learned, this should be solved by proposing a general solution combined linearly with a particular solution $$ W(x,s) = W_c(x,s) + W_p(x,s) $$ I have solved the homogeneous problem to $$ W_c(x,s) =C_1\exp(\sqrt(s)x)+C_2\exp(-\sqrt(s)x) $$ but how can I propose a particular solution?
Thanks in adance.