Using Laplace Transforms to Solve Initial Value Problems Bernd Schr - PowerPoint PPT Presentation

Overview An Example Double Check Using Laplace Transforms to Solve Initial Value Problems Bernd Schr¨ oder logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Using Laplace Transforms to Solve Initial Value Problems

Overview An Example Double Check How Laplace Transforms Turn Initial Value Problems Into Algebraic Equations logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Using Laplace Transforms to Solve Initial Value Problems

Overview An Example Double Check How Laplace Transforms Turn Initial Value Problems Into Algebraic Equations Time Domain ( t ) logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Using Laplace Transforms to Solve Initial Value Problems

Overview An Example Double Check How Laplace Transforms Turn Initial Value Problems Into Algebraic Equations Time Domain ( t ) Original DE & IVP logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Using Laplace Transforms to Solve Initial Value Problems

Overview An Example Double Check How Laplace Transforms Turn Initial Value Problems Into Algebraic Equations Time Domain ( t ) Original L ✲ DE & IVP logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Using Laplace Transforms to Solve Initial Value Problems

Overview An Example Double Check How Laplace Transforms Turn Initial Value Problems Into Algebraic Equations Time Domain ( t ) Original L Algebraic equation for ✲ DE & IVP the Laplace transform logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Using Laplace Transforms to Solve Initial Value Problems

Overview An Example Double Check How Laplace Transforms Turn Initial Value Problems Into Algebraic Equations Time Domain ( t ) Transform domain ( s ) Original L Algebraic equation for ✲ DE & IVP the Laplace transform logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Using Laplace Transforms to Solve Initial Value Problems

Overview An Example Double Check How Laplace Transforms Turn Initial Value Problems Into Algebraic Equations Time Domain ( t ) Transform domain ( s ) Original L Algebraic equation for ✲ DE & IVP the Laplace transform Algebraic solution, partial fractions ❄ logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Using Laplace Transforms to Solve Initial Value Problems

Overview An Example Double Check How Laplace Transforms Turn Initial Value Problems Into Algebraic Equations Time Domain ( t ) Transform domain ( s ) Original L 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 Using Laplace Transforms to Solve Initial Value Problems

Overview An Example Double Check How Laplace Transforms Turn Initial Value Problems Into Algebraic Equations Time Domain ( t ) Transform domain ( s ) Original L 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 Using Laplace Transforms to Solve Initial Value Problems

Overview An Example Double Check How Laplace Transforms Turn Initial Value Problems Into Algebraic Equations Time Domain ( t ) Transform domain ( s ) Original L 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 Using Laplace Transforms to Solve Initial Value Problems

Overview An Example Double Check How Laplace Transforms Turn Initial Value Problems Into Algebraic Equations logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Using Laplace Transforms to Solve Initial Value Problems

Overview An Example Double Check How Laplace Transforms Turn Initial Value Problems Into Algebraic Equations 1. The first key property of the Laplace transform is the way derivatives are transformed. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Using Laplace Transforms to Solve Initial Value Problems

Overview An Example Double Check How Laplace Transforms Turn Initial Value Problems Into Algebraic Equations 1. The first key property of the Laplace transform is the way derivatives are transformed. 1.1 L { y } ( s ) = : Y ( s ) logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Using Laplace Transforms to Solve Initial Value Problems

Overview An Example Double Check How Laplace Transforms Turn Initial Value Problems Into Algebraic Equations 1. The first key property of the Laplace transform is the way derivatives are transformed. 1.1 L { y } ( s ) = : Y ( s ) (This is just notation.) logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Using Laplace Transforms to Solve Initial Value Problems

Overview An Example Double Check How Laplace Transforms Turn Initial Value Problems Into Algebraic Equations 1. The first key property of the Laplace transform is the way derivatives are transformed. 1.1 L { y } ( s ) = : Y ( s ) (This is just notation.) y ′ � � 1.2 L ( s ) = sY ( s ) − y ( 0 ) logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Using Laplace Transforms to Solve Initial Value Problems

Overview An Example Double Check How Laplace Transforms Turn Initial Value Problems Into Algebraic Equations 1. The first key property of the Laplace transform is the way derivatives are transformed. 1.1 L { y } ( s ) = : Y ( s ) (This is just notation.) y ′ � � 1.2 L ( s ) = sY ( s ) − y ( 0 ) y ′′ � ( s ) = s 2 Y ( s ) − sy ( 0 ) − y ′ ( 0 ) � 1.3 L logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Using Laplace Transforms to Solve Initial Value Problems

Overview An Example Double Check How Laplace Transforms Turn Initial Value Problems Into Algebraic Equations 1. The first key property of the Laplace transform is the way derivatives are transformed. 1.1 L { y } ( s ) = : Y ( s ) (This is just notation.) y ′ � � 1.2 L ( s ) = sY ( s ) − y ( 0 ) y ′′ � ( s ) = s 2 Y ( s ) − sy ( 0 ) − y ′ ( 0 ) � 1.3 L � � 1.4 L y ( n ) ( t ) ( s ) = s n Y ( s ) − s n − 1 y ( 0 ) − s n − 2 y ′ ( 0 ) −···− y ( n − 1 ) ( 0 ) logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Using Laplace Transforms to Solve Initial Value Problems

Overview An Example Double Check How Laplace Transforms Turn Initial Value Problems Into Algebraic Equations 1. The first key property of the Laplace transform is the way derivatives are transformed. 1.1 L { y } ( s ) = : Y ( s ) (This is just notation.) y ′ � � 1.2 L ( s ) = sY ( s ) − y ( 0 ) y ′′ � ( s ) = s 2 Y ( s ) − sy ( 0 ) − y ′ ( 0 ) � 1.3 L � � 1.4 L y ( n ) ( t ) ( s ) = s n Y ( s ) − s n − 1 y ( 0 ) − s n − 2 y ′ ( 0 ) −···− y ( n − 1 ) ( 0 ) 2. The right sides above do not involve derivatives of whatever Y is. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Using Laplace Transforms to Solve Initial Value Problems

Overview An Example Double Check How Laplace Transforms Turn Initial Value Problems Into Algebraic Equations 1. The first key property of the Laplace transform is the way derivatives are transformed. 1.1 L { y } ( s ) = : Y ( s ) (This is just notation.) y ′ � � 1.2 L ( s ) = sY ( s ) − y ( 0 ) y ′′ � ( s ) = s 2 Y ( s ) − sy ( 0 ) − y ′ ( 0 ) � 1.3 L � � 1.4 L y ( n ) ( t ) ( s ) = s n Y ( s ) − s n − 1 y ( 0 ) − s n − 2 y ′ ( 0 ) −···− y ( n − 1 ) ( 0 ) 2. The right sides above do not involve derivatives of whatever Y is. 3. The other key property is that constants and sums “factor through” the Laplace transform: logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Using Laplace Transforms to Solve Initial Value Problems

Overview An Example Double Check How Laplace Transforms Turn Initial Value Problems Into Algebraic Equations 1. The first key property of the Laplace transform is the way derivatives are transformed. 1.1 L { y } ( s ) = : Y ( s ) (This is just notation.) y ′ � � 1.2 L ( s ) = sY ( s ) − y ( 0 ) y ′′ � ( s ) = s 2 Y ( s ) − sy ( 0 ) − y ′ ( 0 ) � 1.3 L � � 1.4 L y ( n ) ( t ) ( s ) = s n Y ( s ) − s n − 1 y ( 0 ) − s n − 2 y ′ ( 0 ) −···− y ( n − 1 ) ( 0 ) 2. The right sides above do not involve derivatives of whatever Y is. 3. The other key property is that constants and sums “factor through” the Laplace transform: L { f + g } = L { f } + L { g } and L { af } = a L { f } . logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Using Laplace Transforms to Solve Initial Value Problems