Solving time domain equations in the Laplace domain without inverse transform

Question

Is there a way to obtain a time-domain solution to an equation, posed in the Laplace domain, without first employing the inverse Laplace transform?

As a simple example: $\cos(\omega t) = 0$ has solutions at $t = \frac{\pm0.5\pi + 2 \pi c}{\omega}$, where c is any integer.

How would one go about solving the same equation, with the Laplace transform of $\mathscr L(cos(\omega t)) =\frac{s}{s^2+\omega^2}$ without first employing the inverse Laplace transform?

Albeit this is a much simpler example problem then the problem that one is working on but the solution methodology would be of substantial help. Thank you, in advance.

Answer 1

• $$\cos(\omega t)=0\Longleftrightarrow t=\frac{\pi\left(2\text{n}-1\right)}{2\omega}$$ Where $\omega\ne0$ and $\text{n}\in\mathbb{Z}$

Using Laplace transform:

1. When $\left|\Im\left[\omega\right]\right|<\Re\left[\text{s}\right]$: $$\mathcal{L}_t\left[\cos(\omega t)\right]_{(\text{s})}=\int_0^\infty\cos(\omega t)e^{-\text{s}t}\space\text{d}t=\frac{\text{s}}{\text{s}^2+\omega^2}$$
2. $$\mathcal{L}_t\left[0\right]_{(\text{s})}=\int_0^\infty0e^{-\text{s}t}\space\text{d}t=0$$

So, when $\omega\ne0$:

$$\mathcal{L}_t\left[\cos(\omega t)\right]_{(\text{s})}=\mathcal{L}_t\left[0\right]_{(\text{s})}\Longleftrightarrow\frac{\text{s}}{\text{s}^2+\omega^2}=0\Longleftrightarrow\text{s}=0$$

So now it is not possible as you see. And notice that the inverse Laplace transform of $\text{s}$ equals $\delta'(t)$