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# 6: The Laplace Transform

• • Contributed by Jiří Lebl
• Assistant Professor (Mathematics) at Oklahoma State University
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The Laplace transform can also be used to solve differential equations and reduces a linear differential equation to an algebraic equation, which can then be solved by the formal rules of algebra.

• 6.1: The Laplace Transform
The Laplace transform turns out to be a very efficient method to solve certain ODE problems. In particular, the transform can take a differential equation and turn it into an algebraic equation. If the algebraic equation can be solved, applying the inverse transform gives us our desired solution.
• 6.2: Transforms of derivatives and ODEs
The procedure for linear constant coefficient equations is as follows. We take an ordinary differential equation in the time variable t . We apply the Laplace transform to transform the equation into an algebraic (non differential) equation in the frequency domain.  We solve the equation for X(s) . Then taking the inverse transform, if possible, we find x(t). Unfortunately, not every function has a Laplace transform, not every equation can be solved in this manner.
• 6.3: Convolution
The Laplace transformation of a product is not the product of the transforms. Instead, we introduce the convolution of two functions of t to generate another function of t.
• 6.4: Dirac delta and impulse response
Often in applications we study a physical system by putting in a short pulse and then seeing what the system does. The resulting behavior is often called impulse response.
• 6.E: The Laplace Transform (Exercises)
These are homework exercises to accompany Libl's "Differential Equations for Engineering" Textmap. This is a textbook targeted for a one semester first course on differential equations, aimed at engineering students. Prerequisite for the course is the basic calculus sequence.