Suppose I have an equation like:
$\sin t=t\cos t$
If I take the Laplace transform of both sides I obtain:
$\frac{1}{s^2+1}=\frac{s^2-1}{(s^2+1)^2}$
Simplifying I obtain:
$\frac{s^2-1}{s^2+1}=1$
Taking the inverse laplace transform:
$ \delta(t)-2 \sin t = \delta(t) $
$ \sin t = 0 $
Apart from the solution $t=0$, this method does not yield the correct solution.
So my question is this:
Is this method a valid way to approach the problem, or is there a fundamental flaw with this approach?