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Unformatted text preview: Impulsive Functions Often in applications, we want to solve the differential equation y 00 + p ( x ) y + q ( x ) y = g ( x ) but all we know about g ( x ) is (1) g ( x ) = 0 except on a small interval x ≤ x ≤ x 1 (2) the integral of g ( x ) over this small interval has some known value I . Such functions g ( x ) are called impulsive functions . Method of Dirac Physicist Paul Dirac developed a method for solving such differential equations which was later placed on a firm mathematical footing. The method involves replacing the unknown impulsive function g ( x ) with a very specific formal object called the Dirac Delta function. Definition The Dirac delta function δ ( x- x ) is a formal object that we will consider to be a function with the following properties: δ ( x- x ) = x 6 = x ∞ x = x (1) Z b a f ( x ) δ ( x- x ) = f ( x ) a ≤ x ≤ x otherwise (2) I Example Find the Laplace transform of the Dirac delta function δ ( x- c ) ....
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- Laplace, Sin, Dirac delta function