please if you are not going to answer and instead tell me that I should know this because its 3rd grade math then simply dont respond. I never get straightforward answers on this forum, only sarcastic ones. I am not a math prodigy I dont understand all notation, I just want an answer to the question, not a reality check that I know nothing.
I believe this question can be easily answered by those who know
Thx to those that will answer in a straightforward manner.
In general, the Laplace Transform is usually applied to the Linear Differential Equations, with constant coefficients and non-homogeneous (when $g(x)$ is any function different than a constant, i.e. a function of the denpendent variable, $x$). For instance:
$$y'''(x) - 5y'(x) = g(x)$$
However, we may use the Laplace Transform so as to solve some integro-differential equations (equations with derivatives of $y$ and integrals) as well as some definite and improper integrals. For instance:
$$y''(x) + y'(x) + \int_0^xy(u)du = g(x)$$ $$\int_0^{\pi/2} \frac{\sin(x)}{x}dx = \frac{\pi}{2}$$
That's possible because:
$$\mathcal{L} \left( \int_0^xy(u)du \right) = \frac{\bar{y}(s)}{s}$$