Solutions of differential equations in laplace transform space

The Laplace transform is applied to integrable functions in [0,├ ∞)┤ by L{f}=∫_0^∞▒〖f(x) e^(-sx) dx〗. This transformation reduces differential equations to algebraic equations and is useful for solving an important class of differential equations. Furthermore, the Bilateral Laplace transform is applied to integrable functions defined on the interval (-∞,+∞) and is also used to solve differential equations. In the classical method, when solving differential equations using such transformations, the solution is obtained by taking the Laplace (or Bilateral Laplace) transform of the differential equation. We use a different perspective. We assume that the differential equation itself is in Laplace (Bilateral Laplace) space. So, we assume that the solution y is the Laplace transform of an unknown function in the form y(s)=L{f(x)}(s) (or y(s)=B{f(x)}(s)) and search for solutions accordingly. We use the Laplace transform or Bilateral Laplace transform depending on the conditions and properties of the differential equation.