Homogeneous Systems of Linear Differential Equations with Constant - PowerPoint PPT Presentation

Overview Complex Eigenvalues An Example Homogeneous Systems of Linear Differential Equations with Constant Coefficients and Complex Eigenvalues Bernd Schr¨ oder logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Homogeneous Systems of Linear Differential Equations with Constant Coefficients and Complex Eigenvalues

Overview Complex Eigenvalues An Example Systems of Linear Differential Equations with Constant Coefficients and Complex Eigenvalues logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Homogeneous Systems of Linear Differential Equations with Constant Coefficients and Complex Eigenvalues

Overview Complex Eigenvalues An Example Systems of Linear Differential Equations with Constant Coefficients and Complex Eigenvalues 1. These systems are typically written in matrix form as y ′ = A � � y , where A is an n × n matrix and � y is a column vector with n rows. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Homogeneous Systems of Linear Differential Equations with Constant Coefficients and Complex Eigenvalues

Overview Complex Eigenvalues An Example Systems of Linear Differential Equations with Constant Coefficients and Complex Eigenvalues 1. These systems are typically written in matrix form as y ′ = A � � y , where A is an n × n matrix and � y is a column vector with n rows. 2. The theory guarantees that there will always be a set of n linearly independent solutions { � y 1 ,...,� y n } . logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Homogeneous Systems of Linear Differential Equations with Constant Coefficients and Complex Eigenvalues

Overview Complex Eigenvalues An Example Systems of Linear Differential Equations with Constant Coefficients and Complex Eigenvalues 1. These systems are typically written in matrix form as y ′ = A � � y , where A is an n × n matrix and � y is a column vector with n rows. 2. The theory guarantees that there will always be a set of n linearly independent solutions { � y 1 ,...,� y n } . y = c 1 y 1 + ··· + c n 3. Every solution is of the form � � � y n . logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Homogeneous Systems of Linear Differential Equations with Constant Coefficients and Complex Eigenvalues

Overview Complex Eigenvalues An Example Systems of Linear Differential Equations with Constant Coefficients and Complex Eigenvalues 1. These systems are typically written in matrix form as y ′ = A � � y , where A is an n × n matrix and � y is a column vector with n rows. 2. The theory guarantees that there will always be a set of n linearly independent solutions { � y 1 ,...,� y n } . y = c 1 y 1 + ··· + c n 3. Every solution is of the form � � � y n . x ′ = D 4. If A = Φ D Φ − 1 and � x solves � � x , then A ( Φ x ) � logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Homogeneous Systems of Linear Differential Equations with Constant Coefficients and Complex Eigenvalues

Overview Complex Eigenvalues An Example Systems of Linear Differential Equations with Constant Coefficients and Complex Eigenvalues 1. These systems are typically written in matrix form as y ′ = A � � y , where A is an n × n matrix and � y is a column vector with n rows. 2. The theory guarantees that there will always be a set of n linearly independent solutions { � y 1 ,...,� y n } . y = c 1 y 1 + ··· + c n 3. Every solution is of the form � � � y n . x ′ = D 4. If A = Φ D Φ − 1 and � x solves � � x , then x ) = Φ D Φ − 1 ( Φ A ( Φ x ) � � logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Homogeneous Systems of Linear Differential Equations with Constant Coefficients and Complex Eigenvalues

Overview Complex Eigenvalues An Example Systems of Linear Differential Equations with Constant Coefficients and Complex Eigenvalues 1. These systems are typically written in matrix form as y ′ = A � � y , where A is an n × n matrix and � y is a column vector with n rows. 2. The theory guarantees that there will always be a set of n linearly independent solutions { � y 1 ,...,� y n } . y = c 1 y 1 + ··· + c n 3. Every solution is of the form � � � y n . x ′ = D 4. If A = Φ D Φ − 1 and � x solves � � x , then x ) = Φ D Φ − 1 ( Φ A ( Φ x ) = Φ D � � � x logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Homogeneous Systems of Linear Differential Equations with Constant Coefficients and Complex Eigenvalues

Overview Complex Eigenvalues An Example Systems of Linear Differential Equations with Constant Coefficients and Complex Eigenvalues 1. These systems are typically written in matrix form as y ′ = A � � y , where A is an n × n matrix and � y is a column vector with n rows. 2. The theory guarantees that there will always be a set of n linearly independent solutions { � y 1 ,...,� y n } . y = c 1 y 1 + ··· + c n 3. Every solution is of the form � � � y n . x ′ = D 4. If A = Φ D Φ − 1 and � x solves � � x , then x ) = Φ D Φ − 1 ( Φ x ′ A ( Φ x ) = Φ D x = Φ � � � � logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Homogeneous Systems of Linear Differential Equations with Constant Coefficients and Complex Eigenvalues

Overview Complex Eigenvalues An Example Systems of Linear Differential Equations with Constant Coefficients and Complex Eigenvalues 1. These systems are typically written in matrix form as y ′ = A � � y , where A is an n × n matrix and � y is a column vector with n rows. 2. The theory guarantees that there will always be a set of n linearly independent solutions { � y 1 ,...,� y n } . y = c 1 y 1 + ··· + c n 3. Every solution is of the form � � � y n . x ′ = D 4. If A = Φ D Φ − 1 and � x solves � � x , then x ′ = ( Φ x ) = Φ D Φ − 1 ( Φ x ) ′ , A ( Φ x ) = Φ D x = Φ � � � � � logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Homogeneous Systems of Linear Differential Equations with Constant Coefficients and Complex Eigenvalues

Overview Complex Eigenvalues An Example Systems of Linear Differential Equations with Constant Coefficients and Complex Eigenvalues 1. These systems are typically written in matrix form as y ′ = A � � y , where A is an n × n matrix and � y is a column vector with n rows. 2. The theory guarantees that there will always be a set of n linearly independent solutions { � y 1 ,...,� y n } . y = c 1 y 1 + ··· + c n 3. Every solution is of the form � � � y n . x ′ = D 4. If A = Φ D Φ − 1 and � x solves � � x , then x ′ = ( Φ x ) = Φ D Φ − 1 ( Φ x ) ′ , A ( Φ x ) = Φ D x = Φ � � � � � y ′ = A that is, � y = Φ � x solves � � y . logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Homogeneous Systems of Linear Differential Equations with Constant Coefficients and Complex Eigenvalues

Overview Complex Eigenvalues An Example Systems of Linear Differential Equations with Constant Coefficients and Complex Eigenvalues 1. These systems are typically written in matrix form as y ′ = A � � y , where A is an n × n matrix and � y is a column vector with n rows. 2. The theory guarantees that there will always be a set of n linearly independent solutions { � y 1 ,...,� y n } . y = c 1 y 1 + ··· + c n 3. Every solution is of the form � � � y n . x ′ = D 4. If A = Φ D Φ − 1 and � x solves � � x , then x ′ = ( Φ x ) = Φ D Φ − 1 ( Φ x ) ′ , A ( Φ x ) = Φ D x = Φ � � � � � y ′ = A that is, � y = Φ � x solves � � y . y ′ = A 5. Conversely, every solution of � � y can be obtained as above. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Homogeneous Systems of Linear Differential Equations with Constant Coefficients and Complex Eigenvalues

Overview Complex Eigenvalues An Example Systems of Linear Differential Equations with Constant Coefficients and Complex Eigenvalues logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Homogeneous Systems of Linear Differential Equations with Constant Coefficients and Complex Eigenvalues

Overview Complex Eigenvalues An Example Systems of Linear Differential Equations with Constant Coefficients and Complex Eigenvalues 6. So if we can find a representation A = Φ D Φ − 1 so that x ′ = D y ′ = A � � x is easy to solve, then � � y is also easy to solve. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Homogeneous Systems of Linear Differential Equations with Constant Coefficients and Complex Eigenvalues