Laplace transforms can help simplify and solve complex differential equations by converting them into algebraic equations, which are easier to manipulate and solve.
Some common mistakes include not properly setting up the initial conditions, forgetting to take into account the inverse laplace transform, and not accounting for any discontinuities in the original function.
Laplace transforms are most effective for solving linear differential equations with constant coefficients. They may not be as useful for non-linear or higher order differential equations.
One limitation is that laplace transforms can only be used for equations with continuous functions. They also may not work for equations with varying coefficients or with certain types of boundary conditions.
You can check your solution by taking the inverse laplace transform and comparing it to the original differential equation. You can also plug your solution into the original equation and see if it satisfies the initial conditions.