Dierential Equations & Linear Algebra
Lecture 6 Jianli XIE Email: xjl@sjtu.edu.cn
Department of Mathematics, SJTU
Fall 2010
Contents
Structure of solutions of linear homogeneous equations Existence and uniqueness of solutions Principle of superposition Fu
Dierential Equations & Linear Algebra
Lecture 20 Jianli XIE Email: xjl@sjtu.edu.cn
Department of Mathematics, SJTU
Fall 2010
Contents
Homogeneous linear systems with constant coecients Geometric study: direction eld, phase portrait Case I: real and dieren
Dierential Equations & Linear Algebra
Lecture 21 Jianli XIE Email: xjl@sjtu.edu.cn
Department of Mathematics, SJTU
Fall 2010
Contents
Three main patterns
Contents
Three main patterns
Repeated eigenvalues
Summary of three main cases
For second order system
Dierential Equations & Linear Algebra
Lecture 22 Jianli XIE Email: xjl@sjtu.edu.cn
Department of Mathematics, SJTU
Fall 2010
Contents
Fundamental matrices
Contents
Fundamental matrices Nonhomogeneous linear systems Method of diagonalization Method of unde
Dierential Equations & Linear Algebra
Lecture 24 Jianli XIE Email: xjl@sjtu.edu.cn
Department of Mathematics, SJTU
Fall 2010
Contents
Fourier series for even and odd functions
Contents
Fourier series for even and odd functions
Separation of variables
Four
Dierential Equations & Linear Algebra
Lecture 1 Jianli XIE Email: xjl@sjtu.edu.cn
Department of Mathematics, SJTU
Fall 2010
Contents
Course Syllabus
Contents
Course Syllabus Introduction to Dierential Equations
Contents
Course Syllabus Introduction to Die
Dierential Equations & Linear Algebra
Lecture 2 Jianli XIE Email: xjl@sjtu.edu.cn
Department of Mathematics, SJTU
Fall 2010
Contents
Linear equations: Integrating factors
Contents
Linear equations: Integrating factors Separable equations
Contents
Linear e
Dierential Equations & Linear Algebra
Lecture 3 Jianli XIE Email: xjl@sjtu.edu.cn
Department of Mathematics, SJTU
Fall 2010
Contents
Modeling with rst order ODE
Contents
Modeling with rst order ODE Dierences between linear and nonlinear equations
Contents
Dierential Equations & Linear Algebra
Lecture 19 Jianli XIE Email: xjl@sjtu.edu.cn
Department of Mathematics, SJTU
Fall 2010
Contents
Systems of rst order linear equations General form of systems of rst order equations Existence and uniqueness theorems
Co
Dierential Equations & Linear Algebra
Lecture 18 Jianli XIE Email: xjl@sjtu.edu.cn
Department of Mathematics, SJTU
Fall 2010
Contents
Eigenvalues and eigenvectors
Eigenvalues and eigenvectors
The equation Ax = y, can be viewed as a linear transformation t
Dierential Equations & Linear Algebra
Lecture 16 Jianli XIE Email: xjl@sjtu.edu.cn
Department of Mathematics, SJTU
Fall 2010
Contents
Inner product, norm, orthogonality
Contents
Inner product, norm, orthogonality
Euclidean space Rn
Inner product
We have k
Dierential Equations & Linear Algebra
Lecture 7 Jianli XIE Email: xjl@sjtu.edu.cn
Department of Mathematics, SJTU
Fall 2010
Contents
Structure of solutions of nonhomogeneous equations
Contents
Structure of solutions of nonhomogeneous equations Nonhomogene
Dierential Equations & Linear Algebra
Lecture 8 Jianli XIE Email: xjl@sjtu.edu.cn
Department of Mathematics, SJTU
Fall 2010
Contents
Method of variation of parameters An illustrative example Method of variation of parameters
Contents
Method of variation o
Dierential Equations & Linear Algebra
Lecture 9 Jianli XIE Email: xjl@sjtu.edu.cn
Department of Mathematics, SJTU
Fall 2010
Contents
Determinants and Cramers rule
Contents
Determinants and Cramers rule General theory of nth order linear equations Existenc
Dierential Equations & Linear Algebra
Lecture 10 Jianli XIE Email: xjl@sjtu.edu.cn
Department of Mathematics, SJTU
Fall 2010
Contents
Homogeneous equations with constant coecients Case I: real and unequal roots Case II: complex roots Case III: repeated ro
Dierential Equations & Linear Algebra
Lecture 11 Jianli XIE Email: xjl@sjtu.edu.cn
Department of Mathematics, SJTU
Fall 2010
Contents
Review of power series Radius of convergence Operations of convergent series Taylor series and Maclaurin series Shift of
Dierential Equations & Linear Algebra
Lecture 12 Jianli XIE Email: xjl@sjtu.edu.cn
Department of Mathematics, SJTU
Fall 2010
Contents
Series solutions near an ordinary point
Contents
Series solutions near an ordinary point
Brief review for midterm
Example
Dierential Equations & Linear Algebra
Lecture 13 Jianli XIE Email: xjl@sjtu.edu.cn
Department of Mathematics, SJTU
Fall 2010
Contents
Elementary linear algebra Properties of matrices
Contents
Elementary linear algebra Properties of matrices System of line
Dierential Equations & Linear Algebra
Lecture 14 Jianli XIE Email: xjl@sjtu.edu.cn
Department of Mathematics, SJTU
Fall 2010
Contents
Homogeneous systems
Contents
Homogeneous systems
Non-singular matrices
Example
Example Solve the system 6x3 + 2x4 4x5 8x6
Dierential Equations & Linear Algebra
Lecture 15 Jianli XIE Email: xjl@sjtu.edu.cn
Department of Mathematics, SJTU
Fall 2010
Contents
Vector space
Contents
Vector space Linear dependence and independence
Contents
Vector space Linear dependence and indepen
Dierential Equations & Linear Algebra
Lecture 5 Jianli XIE Email: xjl@sjtu.edu.cn
Department of Mathematics, SJTU
Fall 2010
Contents
Homogeneous equations with constant coecients Homogeneous and nonhomogeneous equations Case I: two real dierent roots Case