MAT3706 - Ordinary Differential
Equations
Summary
Chapter 1 - Linear Systems of DE’s, Elimination- and Triangulation Methods 4
Outcomes 4
[Non]Homogeneous Systems & Matrix Notation 5
Example: 5
Polynomial Differential Operators & Degeneracy 6
Solutions to Systems of DE’s 6
Superposition Principle 6
Linear Independence 7
Existence of Solutions of Homogeneous System (General Solution) 7
Existence of Solutions of Non-Homogeneous System (General Solution) 8
Correct Number of Arbitrary Constants for General Solutions 8
Method of Elimination (Operator Method) 9
Polynomial operators 9
Write a System of DE’s in Operator Notation 9
Example: 9
Operator Method (Method of Elimination) 10
Example: 10
Triangulation Method 11
Degenerate Systems of DEs 13
Degeneracy 13
Determine if No or Infinite Solutions 13
Example: 14
Apply Techniques 15
Chapter 2 - Eigenvalues, Eigenvectors, Systems of Linear Equations with Constant
Coefficients 16
Outcomes 16
Eigenvalue-Eigenvector Method 17
Superposition Principle 18
Linear Independence of Eigenvectors 18
Wronskian 18
Example: 19
Complex Eigenvalues and Eigenvectors 21
New roots of X`=AX 23
Example: 24
Chapter3 - Generalised Eigenvectors (Root Vector) & System of Linear DEs 27
Outcomes 27
Generalised Eigenvectors (Root Vector) 27
Example: 28
Solutions of X`=AX using Root Vectors 30
, Solvability of IVP X`=AX(t) 32
Existence theorem for Linear Systems of DEs with Constant Coefficients 32
Chapter 4 - Fundamental Matrices, Non-holonomic Systems, Inequality of Gronwall 33
Outcomes 33
Fundamental Matrices 33
Example: 33
Uniqueness Theorem: Linear Systems with Constant Coefficients 36
Uniqueness Theorem: Linear Systems with Constant Coefficients 36
Application of the Uniqueness Theorem 37
Variation of Parameters: Non-homogeneous Problem 38
Example: 38
Uniqueness Theorem: Nonhomogeneous IVP 42
Example: 42
Inequality of Gronwall 44
Inequality of Gronwall 45
Example: 45
Example: 45
Growth of Solutions 46
Chapter 5 - Higher Order 1-D Equations as System of First Order Equations 48
Outcomes 48
Companion Systems for Higher Order 1-D DEs 48
Example: 48
Companion matrix 50
Companion System 50
Chapter 6 - Analytical Matrices & Power Series Solutions of Systems of DEs 54
Outcomes 54
Power Series Expansion of Analytical Functions 54
Taylor Series for some Elementary functions 55
Example: 56
Series Solution for Ẋ = A(t)X 57
Exponential of a Matrix 61
Chapter 7 - Nonlinear Systems, Existence & Uniqueness for Linear Systems 62
Outcomes 62
Nonlinear Equations & Systems 62
Nonlinear Equation 62
[Non]-Autonomous Systems 62
Example: 62
Example: 63
Numerical Solutions of DE 64
Existence & Uniqueness of Linear Systems of DEs 64
Chapter 8 - Qualitative Theory of DE, Stability of Solutions of Linear Systems,
Linearization of Nonlinear Systems 65
Outcomes 65
Plane Autonomous System 65
, Finding Critical Points 65
Example: 65
Changing to Polar Coordinates 66
Example: 66
Stability of Linear System 67
Linearization & Local Stability 67
Stable Critical Point 68
Unstable Critical Point 69
Example: 69
Linearization 71
Jacobian Matrix 71
Example: 72
Equations
Summary
Chapter 1 - Linear Systems of DE’s, Elimination- and Triangulation Methods 4
Outcomes 4
[Non]Homogeneous Systems & Matrix Notation 5
Example: 5
Polynomial Differential Operators & Degeneracy 6
Solutions to Systems of DE’s 6
Superposition Principle 6
Linear Independence 7
Existence of Solutions of Homogeneous System (General Solution) 7
Existence of Solutions of Non-Homogeneous System (General Solution) 8
Correct Number of Arbitrary Constants for General Solutions 8
Method of Elimination (Operator Method) 9
Polynomial operators 9
Write a System of DE’s in Operator Notation 9
Example: 9
Operator Method (Method of Elimination) 10
Example: 10
Triangulation Method 11
Degenerate Systems of DEs 13
Degeneracy 13
Determine if No or Infinite Solutions 13
Example: 14
Apply Techniques 15
Chapter 2 - Eigenvalues, Eigenvectors, Systems of Linear Equations with Constant
Coefficients 16
Outcomes 16
Eigenvalue-Eigenvector Method 17
Superposition Principle 18
Linear Independence of Eigenvectors 18
Wronskian 18
Example: 19
Complex Eigenvalues and Eigenvectors 21
New roots of X`=AX 23
Example: 24
Chapter3 - Generalised Eigenvectors (Root Vector) & System of Linear DEs 27
Outcomes 27
Generalised Eigenvectors (Root Vector) 27
Example: 28
Solutions of X`=AX using Root Vectors 30
, Solvability of IVP X`=AX(t) 32
Existence theorem for Linear Systems of DEs with Constant Coefficients 32
Chapter 4 - Fundamental Matrices, Non-holonomic Systems, Inequality of Gronwall 33
Outcomes 33
Fundamental Matrices 33
Example: 33
Uniqueness Theorem: Linear Systems with Constant Coefficients 36
Uniqueness Theorem: Linear Systems with Constant Coefficients 36
Application of the Uniqueness Theorem 37
Variation of Parameters: Non-homogeneous Problem 38
Example: 38
Uniqueness Theorem: Nonhomogeneous IVP 42
Example: 42
Inequality of Gronwall 44
Inequality of Gronwall 45
Example: 45
Example: 45
Growth of Solutions 46
Chapter 5 - Higher Order 1-D Equations as System of First Order Equations 48
Outcomes 48
Companion Systems for Higher Order 1-D DEs 48
Example: 48
Companion matrix 50
Companion System 50
Chapter 6 - Analytical Matrices & Power Series Solutions of Systems of DEs 54
Outcomes 54
Power Series Expansion of Analytical Functions 54
Taylor Series for some Elementary functions 55
Example: 56
Series Solution for Ẋ = A(t)X 57
Exponential of a Matrix 61
Chapter 7 - Nonlinear Systems, Existence & Uniqueness for Linear Systems 62
Outcomes 62
Nonlinear Equations & Systems 62
Nonlinear Equation 62
[Non]-Autonomous Systems 62
Example: 62
Example: 63
Numerical Solutions of DE 64
Existence & Uniqueness of Linear Systems of DEs 64
Chapter 8 - Qualitative Theory of DE, Stability of Solutions of Linear Systems,
Linearization of Nonlinear Systems 65
Outcomes 65
Plane Autonomous System 65
, Finding Critical Points 65
Example: 65
Changing to Polar Coordinates 66
Example: 66
Stability of Linear System 67
Linearization & Local Stability 67
Stable Critical Point 68
Unstable Critical Point 69
Example: 69
Linearization 71
Jacobian Matrix 71
Example: 72
