Laplace Transformation Applications

Question

In one of our Mathematics lecture our Prof told us that similar to Logarithmic Transformations we can use Laplace Transformations to solve difficult equations.

What kind of equations do Laplace transformations help to solve ?

Answer 1

Linear constant-coefficient ordinary and partial differential equations, mainly (for the partial differential equation case, the coefficients are constant in the "time" variable, not necessarily the space variable). Also some integral equations.

Answer 2

Laplace Equations really shine when you start dealing with differential equations that have discontinuous or otherwise erratic behavior in them. Example: Lets say you have a loaded spring set in motion (based off some IVP) on an elevator and you want to model how it moves as you go from floor 1 to 100, but the elevator stops intermittently at various floors (at which point there is no driving force) or it is rising or dropping to various stories until it reaches the 100th floor. I can think of it like I'm turning a functino on and off at various times t, and this is modelled using the heaviside function, which in order to solve ODE with heaviside functions in it you most likely will need to use Laplace transformations.

Another example: I have a spring that can move up or down, and i "randomly" (you know when, but its not model-able by any single 'function') strike it with a hammer. It can be thought of like the hammer imparts a nearly infinite force for a infinitesimally small period, creating an infinite discontinuity, but the amount of energy that strike imparts is finite. This is modeled with the Dirac-delta (or impulse) function, which once again is a problem where Laplace works well for it.

Essentially since we are transforming our equations with an integral (Riemann sum) transform we can get away with dealing with non-differentiable points in our function without the equation exploding in our face.