Laplace Transformation

The Laplace Transformation
Definition
The Laplace transform converts integral and differential equations into algebraic equations. Laplace transforms are invaluable for any engineer’s mathematical operation as they make solving linear ODEs and related initial value problems, as well as systems of linear ODEs, much easier.

The key motivation for learning about Laplace transforms is that the process of solving an ODE is simplified to an algebraic problem (and transformations). The Laplace transform method has two main advantages over the other methods:
Problems are solved more directly: Initial value problems are solved without first determining a general solution. Nonhomogeneous ODEs are solved without first solving the corresponding homogeneous ODE.
More importantly, the use of the unit step function and Dirac’s delta make the method particularly powerful for problems with inputs (driving forces) that have discontinuities or represent short impulses or complicated periodic functions.

The Laplace transform L

We’ll be interested in functions defined for t?0.
The Laplace transform of a function f is the function F=L(f) defined by:
F(s)=L f(t)=?_0^???f(t) e^(-st) dt?

F is a complex-valued function of complex numbers
s is called the (complex) frequency variable, s*t is unitless
Lower case letter denotes function; capital letter denotes its Laplace transform, e.g., U denotes L(u), V_in denotes L(V_in), etc.
Note that the Laplace transform is called an integral transform because it transforms (changes) a function in one space to a function in another space by a process of integration that involves a kernel (e^(-st)). The kernel or kernel function is a function of the variables in the two spaces and defines the integral transform.