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Tests

Test preparation and submissions: You must read and understand the policy about cheating in this course given in Student Honor Code of  the syllabus.  It will be strictly enforced.  All tests (online or in-person) will be conducted on Mondays, 7-10 pm, except possibly the final or if Monday is a holiday. A place for in-person tests will be posted below. All tests are free response. Write the problem number, then your solution with all technical details and reasoning (e.g., stating the concept that is used), box your final answer.  Then proceed to a next problem. Problems can be solved in any order. There will be no credit for answers without technical details and reasoning. At the bottom of the last page write and sign the student honesty pledge. It is highly appreciated if you try to keep your handwriting neat and clear. Enumerate all pages as 1/N, 2/N, etc, where N=the total number of pages. For submission via Canvas, scan your work and combine it into a single PDF file in the order of increasing page numbers. The file name must have the form: LastNameFirstNameExamM where M = the test number. Then upload your work via Canvas. Make sure that you don’t use the highest resolution which could make your file about 100 Meg, a reasonable size is 2-10 Meg. Large files require a VERY long time to download and upload via Canvas. Late submissions or submissions without signed student honesty pledge will not be accepted.
Use of computational software in online tests: You may use computers to check your answers but you will not get credit for an answer not supported by technical details with a clear indication which concept was used and how it was used. For example, giving eigenvectors and eigenvalues of a matrix with supporting arguments like “Matlab gives that” yields NO full credit. You have to find and solve the equation for eigenvalues and then solve the corresponding linear systems to find eigenvectors, presenting technical details of the process. All intermediate technical problems from linear algebra must be supported by technical details when solving any test problems. This policy will be strictly enforced.

Test 1 is on May 20, 7-9 pm, via Canvas; the test covers Lectures 1-7
Test 1 with Solutions
Test 2 is on May 28, 7-9 pm, via Canvas; the test covers Lectures 8-12;
Test 2 with Solutions
Test 3 is on June 3, in-person; place and time to be posted; the test covers Lectures 10-16;
Test 3 with Solutions
Test 4 is on June 10, 7-9-pm, via Canvas; the test covers Lectures 17-21;
Test 4 with Solutions
Test 5 is on June 17, 6-8 pm, via Canvas; the test covers Lectures 23-25;
Test 5 with Solutions
Test 6 (final) is on June 21, in-person, FAC 127, 2 pm; the exam covers Topics 3 and 4

Lectures

This is a preliminary schedule. The topics will be updated as the course progresses. Video lectures are NOT identical to the actual lectures in the classroom and can be viewed as a supplementary material.

Topic 1: Review (week 1, May 13-17)

Lecture 1: Complex numbers (44 min, not covered in class).
Lecture 2: Functions of a complex variable (56 min, not covered in class).
Lecture 3: Linear differential equations with constant coefficients operators (67 min). Homogeneous linear differential equations with constant coefficients. Characteristic equation. Real and complex roots. Linearly independent solutions. General solution. Additional reading: Section 6.2 (at least, read all examples).  For second order equations you might want to review Sections 4.2, 4.3. Homework: Section 6.2: 3, 5, 9, 11, 15, 17, 19, 31
Lecture 4: Non-Homogeneous linear differential equations with constant coefficients (84 min). The method of undetermined coefficients. Particular solution. General solution. The initial value problem. Additional reading: Section 6.3 (at least, read all examples).  For second order equations you might want to review Section 4.4. Homework: Section 6.3: 1, 3, 5, 9, 13, 17, 19, 25, 31
Lecture 5: Justification of the method of undetermined coefficients (55 min). The complex and real approaches. Examples of higher order equations. Homework: the same as for Lecture 4
Lecture 6: The initial value problem for linear differential equations (79 min). Homework: the same as for Lecture 4
Lecture 7: Systems of linear differential equations (73 min). The elimination method. The method of Laplace transform. Additional reading: The elimination method (Section 5.2). Laplace transform method (Section 7.10) Homework: Section 7.10: 9, 15, 17, 19; Section 5.2: 7, 19, 25 (you might want to solve the problems in either section by both methods like in the lecture and compare them).
Additional videos with examples: Example for Lecture 3; Example for Lecture 4; Example for Lecture 7 Part 1, Part 2

Topic 2: Systems of first-order linear differential equations (week 2, May 20-24)

Lecture 8: (76 min) Linear differential equations and systems of linear differential equations as a system of first-order linear differential equations (Sec. 9.1). Review of algebra of matrices: addition, multiplication, multiplication by a number, transposed and adjoint matrix, the determinant of a square matrix (Sec. 9.3). Homework: Section 9.1: 5, 7, 9, 11, 13; Section 9.3: 5, 8, 23, 25, 27, 29
Lecture 9: Review of basic concepts of linear algebra (84 min). The dot product in matrix notations. The inverse matrix. Gauss elimination method to find the inverse matrix. The necessary and sufficient criteria for the existence of the inverse matrix. Linear systems. The existence and uniqueness of a solution. General solution. The Fredholm alternative for a system of linear equations. Additional reading: Secs. 9.2 and 9.3. Homework: Sec. 9.3: 9, 11, 13, 16, 19, 30; Sec. 9.2: 10, 11, 13
Lecture 10: Systems of first-order linear differential equations. Calculus with matrices. The existence and uniqueness of a solution to a system of first-order linear differential equations. Linear dependence of vector functions on an interval. Wronskian of a finite set of vector functions. General solution to a homogeneous linear system. General solution to a system of first-order linear differential equations. Additional reading: Sec. 9.4. Homework: Sec. 9.4: 3, 13, 15, 17, 23, 25, 26, 29, 37
Lecture 11: Homogeneous 1st order linear systems. The case of real simple eigenvalues and the case of symmetric matrices. The eigenvalue problem. Linearly independent eigenvectors. General solution the homogeneous system. Additional reading: Sec. 9.5. Homework: Sec. 9.5: 3, 7, 11, 15, 25, 31, 33, 35
Lecture 12: Homogeneous 1st order linear systems. The case of simple complex eigenvalues. Complex eigenvectors of a real matrix. Linearly independent complex-valued solutions corresponding to complex eigenvalues. Real-valued linearly independent solutions corresponding to complex eigenvalues. Additional reading: Sec. 9.6. Homework: Sec. 9.6: 1, 3, 13, 14
Additional videos with examples: Example for the Fredholm alternative (Lecture 9); Example for Lecture 12

Topic 2: Systems of linear first-order differential equations (continued) (week 3, May 28-31)

Lecture 13: Fundamental matrix for a linear system. Representation of a general solution to a homogeneous equation via a fundamental matrix. Solution to the initial value problem via a fundamental matrix. Exponential of a matrix. Definition via a power series. Convergence of the series. Exponential of a diagonalizable matrix. Solution to a homogeneous linear system via the exponential of a matrix. Additional reading: Sec. 9.8. Homework: Sec. 9.8: 3, 5, 7, 9, 12 , 11, 21, 22. (it is better to go over Lectures 13 and 14 before doing these problems)
Lecture 14: Exponential of a general matrix. Jordan decomposition theorem. Jordan standard block of each eigenvalue of a matrix. Exponential of a Jordan block. Nilpotent matrices. Generalized eigenvalue problem for a matrix. Finding n linearly independent generalized eigenvectors for an n by n matrix. Additional reading: Sec. 9.8. Homework: Sec. 9.8: 3, 5, 7, 9, 12 , 11, 21, 22.
Lecture 15: Finding a fundamental matrix for a linear system using linearly independent generalized eigenvectors. Comparison of computing the exponential of a matrix and computing a fundamental matrix.  General solution to a non-homogeneous linear system (method of variation of parameters). Additional reading: Sec. 9.8 (about a fundamental matrix) and Sec. 9.7 (about variation of parameters). Homework: Sec. 9.8: 23, 29; Sec. 9.7: 11, 13, 15, 21, 23 (you can use either exp(At) or a fundamental matrix in the method of variation of parameters).
Lecture 16: Finding a particular solution for special inhomogeneities. The method of undetermined coefficients. Additional reading Sec. 9.7 (method of undetermined coefficients). Homework: Sec 9.7: 3, 5, 7, 9, 23
Additional videos: Summary of the Jordan form and exponential of a matrix; Example from Sec. 9.8

Topic 3: The method of Frobenius and Special Functions (week 4, June 3-7 )

Lecture 17: Power series representation of a solution to second-order linear differential equations with analytic coefficients. Analytic functions. Properties of analytic functions. Standard form of a second-order linear differential equation. Regular and singular points. Power series representation of a general solution near a regular point. Radius of convergence.  Initial value problem. Additional reading: Sec. 8.2 (analytic functions), Secs. 8.3 and 8.4 (power series representation of a solution near a regular point).  Homework: Sec. 8.3: 1, 3, 7, 13, 17, 23; Sec. 8.4: 3, 5, 9, 15, 19
Lecture 18: Method of Frobenius (Part 1). A power series representation of a general solution for a non-homogeneous equation with analytic coefficients. Solutions near a singular point. Review: the Cauchy-Euler equation and its general solution. Classification of singular points of a linear equation. Regular and irregular singular points. Indicial equation. Power series representation of a solution near a regular singular point corresponding  to the larger root of the indicial equation. The theorem of Frobenius. Example: a regular solution to the Bessel equation. Additional reading: Sec. 8.4 (P. 449, a non-homogeneous equation); Sec 8.5 (Cauchy-Euler equation), Sec. 8.6 (Method of Frobenius), Sec 8.8 (Bessel’s equation). Homework: Sec. 8.4: 23, 25; Sec 8.5: 3, 11, 15; Sec 8.6: 3, 5, 9, 11, 17, 21, 25, 29, 31
Lecture 19: Method of Frobenius (Part 2). Second linearly independent solution in the case when the difference of the roots of the indicial equation is not integer. Example: Two linearly independent solutions to Bessel’s equation (when nu is not half-integer). Second linearly independent solution in the case when  the roots of the indicial equation are identical. Example:  Two linearly independent solutions to Bessel’s equation (when nu is zero). A shift of summation index in a power series and its use in finding power series representations of two linearly independent solutions (illustrated with an example). Additional reading: Sec. 8.7 (up to Example 3). Homework: Sec. 8.7: 1, 5, 7, 9, 13, 15,  21 (some of these problems require Lecture 20).
Lecture 20: Method of Frobenius (Part 3). Second linearly independent solution in the case when the difference of the roots of the indicial equation is a non-zero integer. Examples: Two linearly independent solutions to Bessel’s equation of a half-integer order. Two linearly independent solutions to Bessel’s equation of positive integer order.   Additional reading: Sec. 8.7 (Examples 3 and 4). Homework: Sec. 8.7: 1, 5, 7, 9, 13, 15,  21, 26.
Lecture 21: Gamma function. Bessel’s functions of the first kind. Bessel functions of the second kind or Neumann or Weber functions. Hankel’s functions of the first and second kind. Properties of the Bessel functions. Explicit form of spherical Bessel functions. Asymptotic properties of Bessel’s functions for large and small arguments. Additional reading: Sec. 8.8 (Bessel’s equation). Homework: Sec. 8.8: 13, 17, 19, 21, 25, 27

Topic 4: Boundary value problem (week 5, June 10-14)

Lecture 22: Hypergeometric equation. Two linearly independent solutions via the Frobenius method. Gaussian hypergeometric function as a regular solution to the hypergeometric equation. Polynomial solutions to the hypergeometric equation. Legendre equation and its reduction to the hypergeometric equation. Legendre polynomials as polynomial solutions to the Legendre equation. The orthogonality of Legendre polynomials in the interval (-1,1). Other orthogonal polynomials. Additional reading: Sec. 8.8. Homework: Sec. 8.8: 1, 3, 5, 7, 8, 9, 31, 37, 39
Lecture 23: Boundary value problem. Types of boundary conditions (Dirichlet, Neumann, mixed, and periodic). Three basic equations in partial derivatives: Heat equation; wave equation, and Laplace equation in two variables. Initial and boundary conditions. The method of separation of variables. Reduction to a boundary value problem for a second-order differential equation. Boundary value problem for a Cauchy-Euler equation. Additional reading: Secs: 11.1, 11.2, and Sec. 10.1 (derivation of the heat equation), Sec. 10.2 (separation of variables in the heat equation). Homework: Sec. 11.2: 1, 3, 5, 7, 9, 11. 15, 23, 25
Lecture 24: The eigenvalue problem for a differential operator. The Sturm-Liouville problem. Dirichlet, Neumann, mixed, and periodic boundary conditions. Lagrange identity for the Sturm-Liouville operator. Inner product functional spaces. Orthogonality of functions. Symmetric differential operators. Eigenvalues of a symmetric differential operator. Eigenfunctions of symmetric differential operators. Additional reading: Sec. 11.3. Homework: Sec. 11.3: 3, 5, 7, 9, 11, 13
Lecture 25: General method for solving the regular Sturm-Liouville problem. Examples. Expansions over eigenfunctions of a regular Sturm-Liouville operator. Basics theorems about the convergence of the Fourier series. Periodic Sturm-Liouville problem. Additional reading: Sec. 11.3. Homework: Sec. 11.3: 17, 19, 21, 23
Lecture 26: The non-homogeneous problem for a regular Sturm-Liouville operator. The method of expansion over orthonormal eigenfunctions. The existence of a solution. Formal Fourier series for a solution. Additional Reading: Sec. 11.5. Homework: Sec. 11.5: 1, 3, 7, 9, 11, 13 (in Problems 9, 11, 13, assume that f(x) is such that a solution exists)

Topic 4: Boundary value problem (continued) (week 6, June 17-21)

Lecture 27: Non-homogeneous boundary value problems and the Fredholm alternative.  The formal adjoint of a differential operator.  The adjoint boundary value problem. The Fredholm alternative. Additional reading: Sec. 11.4. Homework: Sec. 11.4: 7, 9, 11, 13, 15, 17, 19, 21, 25, 29
Lecture 28: Green’s functions of a differential operator. How to construct Green’s function for a regular Sturm-Liouville operator. Solving a non-homogeneous boundary value problem using Green’s functions. Additional Reading: Sec. 11.6. Homework: Sec.11.6: 3, 5, 7, 11, 15, 17, 19
Lecture 29: Green’s function for a non-symmetric differential operator. Separation of variables in polar coordinates in the Laplace equation. Expansion of a solution into a formal trigonometric Fourier series. Additional reading: Sec. 10.7 (Example 2). Homework: Sec. 11.6: 22, 23, 25; Sec. 10.7: 8, 11, 13
Lecture 30: Applications of the eigenvalue problem for a Sturm-Liouville operator to basic wave and heat equations. Additional reading: Secs: 10.5 (heat equation); Sec. 10.6 (wave equation). Homework: Sec. 10.5: 3, 5, 7, 9; Sec. 10.6: 1, 3, 5, 7, 9