# CHAPTER 6

# Laplace Transforms

Laplace transforms are invaluable for any engineer's mathematical toolbox as they make solving linear ODEs and related initial value problems, as well as systems of linear ODEs, much easier. Applications abound: electrical networks, springs, mixing problems, signal processing, and other areas of engineering and physics.

The process of solving an ODE using the Laplace transform method consists of three steps, shown schematically in Fig. 113:

** Step 1.** The given ODE is transformed into an algebraic equation, called the

**subsidiary equation**.

** Step 2.** The subsidiary equation is solved by purely algebraic manipulations.

** Step 3.** The solution in Step 2 is transformed back, resulting in the solution of the given problem.

The key motivation for learning about Laplace transforms is that the process of solving an ODE is simplified to an algebraic problem (and transformations). This type of mathematics that converts problems of calculus to algebraic problems is known as *operational calculus*. The Laplace transform method has two main advantages over the methods discussed in Chaps. 1–4:

**I.** Problems are solved more directly: Initial value problems are solved without first determining a general solution. Nonhomogenous ODEs are ...

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