Lecture 5.5: Impulse functions Matthew Macauley Department of - PowerPoint PPT Presentation

Lecture 5.5: Impulse functions Matthew Macauley Department of Mathematical Sciences Clemson University http://www.math.clemson.edu/~macaule/ Math 2080, Differential Equations M. Macauley (Clemson) Lecture 5.5: Impulse functions Differential Equations 1 / 4

Motivation Question What is the derivative of the Heavyside function? M. Macauley (Clemson) Lecture 5.5: Impulse functions Differential Equations 2 / 4

The Laplace transforms of a periodic function Example � 0 , t � = p The Dirac delta function centered at p is δ p ( t ) = lim ǫ → 0 δ ǫ p ( t ) = ∞ , t = p Properties � ∞ (i) δ p ( t ) dt = 1. −∞ (ii) L{ δ 0 ( t ) } = 1. M. Macauley (Clemson) Lecture 5.5: Impulse functions Differential Equations 3 / 4

Differential equations with impulse forcing terms Example Solve the IVP: y ′′ + 2 y ′ + 2 y = δ 0 ( t ), y (0) = 0, y ′ (0) = 0. M. Macauley (Clemson) Lecture 5.5: Impulse functions Differential Equations 4 / 4