Laplace transform of $\sin(x(t))$

Question

How to find the Laplace transform of $\sin(x(t))$. Laplace transform of $\sin(t)$ seems to be a simple $1/(1+s^2)$, but how can I solve for this $\sin(x(t))$?

Answer 1

I find it highly unlikely that we can find the Laplace transform of $\sin(x(t))$ in any nice, neat formula. But given a some properties of $x(t)$, we might be able to produce some functional equations for the Laplace transform $L(s)$ of $\sin(x(t))$.

For example, suppose that $x$ satisfies the differential equation $$x'(t)=\tan(x(t))$$ If this is the case, then you can verify by integration by parts that $$L(s)=\frac{\sin(x(0))}{s-1}$$ As a more general example, if $x$ satisfies the differential equation $$x'(t)\cos(x(t))=f(t)$$ then it can be verified (again, by IBP) that $$L(s)=\frac{\sin(x(0))+(\mathcal{L}f)(s)}{s}$$

But, again, I don't think you'll find any nice formula.