Laplace Transforms of Step Functions Bernd Schr oder logo1 Bernd - PowerPoint PPT Presentation

Transforms and New Formulas A Model The Initial Value Problem Double Check Laplace Transforms of Step Functions Bernd Schr¨ oder logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Laplace Transforms of Step Functions

Transforms and New Formulas A Model The Initial Value Problem Double Check Everything Remains As It Was logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Laplace Transforms of Step Functions

Transforms and New Formulas A Model The Initial Value Problem Double Check Everything Remains As It Was No matter what functions arise, the idea for solving differential equations with Laplace transforms stays the same. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Laplace Transforms of Step Functions

Transforms and New Formulas A Model The Initial Value Problem Double Check Everything Remains As It Was No matter what functions arise, the idea for solving differential equations with Laplace transforms stays the same. Time Domain ( t ) logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Laplace Transforms of Step Functions

Transforms and New Formulas A Model The Initial Value Problem Double Check Everything Remains As It Was No matter what functions arise, the idea for solving differential equations with Laplace transforms stays the same. Time Domain ( t ) Original DE & IVP logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Laplace Transforms of Step Functions

Transforms and New Formulas A Model The Initial Value Problem Double Check Everything Remains As It Was No matter what functions arise, the idea for solving differential equations with Laplace transforms stays the same. Time Domain ( t ) L Original ✲ DE & IVP logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Laplace Transforms of Step Functions

Transforms and New Formulas A Model The Initial Value Problem Double Check Everything Remains As It Was No matter what functions arise, the idea for solving differential equations with Laplace transforms stays the same. Time Domain ( t ) L Original ✲ Algebraic equation for DE & IVP the Laplace transform logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Laplace Transforms of Step Functions

Transforms and New Formulas A Model The Initial Value Problem Double Check Everything Remains As It Was No matter what functions arise, the idea for solving differential equations with Laplace transforms stays the same. Time Domain ( t ) Transform domain ( s ) L Original ✲ Algebraic equation for DE & IVP the Laplace transform logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Laplace Transforms of Step Functions

Transforms and New Formulas A Model The Initial Value Problem Double Check Everything Remains As It Was No matter what functions arise, the idea for solving differential equations with Laplace transforms stays the same. Time Domain ( t ) Transform domain ( s ) L Original ✲ Algebraic equation for DE & IVP the Laplace transform Algebraic solution, partial fractions ❄ logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Laplace Transforms of Step Functions

Transforms and New Formulas A Model The Initial Value Problem Double Check Everything Remains As It Was No matter what functions arise, the idea for solving differential equations with Laplace transforms stays the same. Time Domain ( t ) Transform domain ( s ) L Original ✲ Algebraic equation for DE & IVP the Laplace transform Algebraic solution, partial fractions ❄ Laplace transform of the solution logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Laplace Transforms of Step Functions

Transforms and New Formulas A Model The Initial Value Problem Double Check Everything Remains As It Was No matter what functions arise, the idea for solving differential equations with Laplace transforms stays the same. Time Domain ( t ) Transform domain ( s ) L Original ✲ Algebraic equation for DE & IVP the Laplace transform Algebraic solution, partial fractions ❄ L − 1 Laplace transform ✛ of the solution logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Laplace Transforms of Step Functions

Transforms and New Formulas A Model The Initial Value Problem Double Check Everything Remains As It Was No matter what functions arise, the idea for solving differential equations with Laplace transforms stays the same. Time Domain ( t ) Transform domain ( s ) L Original ✲ Algebraic equation for DE & IVP the Laplace transform Algebraic solution, partial fractions ❄ L − 1 Laplace transform ✛ Solution of the solution logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Laplace Transforms of Step Functions

Transforms and New Formulas A Model The Initial Value Problem Double Check The Unit Step Function logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Laplace Transforms of Step Functions

Transforms and New Formulas A Model The Initial Value Problem Double Check The Unit Step Function � 1; for t ≥ 0 , 1. U ( t ) = 0; for t < 0 , logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Laplace Transforms of Step Functions

Transforms and New Formulas A Model The Initial Value Problem Double Check The Unit Step Function � 1; for t ≥ 0 , 1. U ( t ) = 0; for t < 0 , logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Laplace Transforms of Step Functions

Transforms and New Formulas A Model The Initial Value Problem Double Check The Unit Step Function � 1; for t ≥ 0 , 1. U ( t ) = 0; for t < 0 , logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Laplace Transforms of Step Functions

Transforms and New Formulas A Model The Initial Value Problem Double Check The Unit Step Function � 1; for t ≥ 0 , 1. U ( t ) = 0; for t < 0 , 2. The unit step function can be shifted and then used to model the switching on and off of another function. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Laplace Transforms of Step Functions

Transforms and New Formulas A Model The Initial Value Problem Double Check The Unit Step Function � 1; for t ≥ 0 , 1. U ( t ) = 0; for t < 0 , 2. The unit step function can be shifted and then used to model the switching on and off of another function. 3. The function U ( t − a ) − U ( t − b ) is equal to 1 on [ a , b ) and equal to zero outside [ a , b ) . logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Laplace Transforms of Step Functions

Transforms and New Formulas A Model The Initial Value Problem Double Check The Unit Step Function � 1; for t ≥ 0 , 1. U ( t ) = 0; for t < 0 , 2. The unit step function can be shifted and then used to model the switching on and off of another function. 3. The function U ( t − a ) − U ( t − b ) is equal to 1 on [ a , b ) and equal to zero outside [ a , b ) . logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Laplace Transforms of Step Functions

Transforms and New Formulas A Model The Initial Value Problem Double Check The Unit Step Function � 1; for t ≥ 0 , 1. U ( t ) = 0; for t < 0 , 2. The unit step function can be shifted and then used to model the switching on and off of another function. 3. The function U ( t − a ) − U ( t − b ) is equal to 1 on [ a , b ) and equal to zero outside [ a , b ) . logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Laplace Transforms of Step Functions

Transforms and New Formulas A Model The Initial Value Problem Double Check The Unit Step Function � 1; for t ≥ 0 , 1. U ( t ) = 0; for t < 0 , 2. The unit step function can be shifted and then used to model the switching on and off of another function. 3. The function U ( t − a ) − U ( t − b ) is equal to 1 on [ a , b ) and equal to zero outside [ a , b ) . 4. The function f ( t ) U ( t − a ) − f ( t ) U ( t − b ) is equal to f on [ a , b ) and equal to zero outside [ a , b ) . logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Laplace Transforms of Step Functions

Transforms and New Formulas A Model The Initial Value Problem Double Check The Unit Step Function � 1; for t ≥ 0 , 1. U ( t ) = 0; for t < 0 , 2. The unit step function can be shifted and then used to model the switching on and off of another function. 3. The function U ( t − a ) − U ( t − b ) is equal to 1 on [ a , b ) and equal to zero outside [ a , b ) . 4. The function f ( t ) U ( t − a ) − f ( t ) U ( t − b ) is equal to f on [ a , b ) and equal to zero outside [ a , b ) . ( s ) = e − as F ( s ) . � � 5. L f ( t − a ) U ( t − a ) logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Laplace Transforms of Step Functions