Math 2243: Class Outlines

o 08/06/04: Final exam. Answers: 1. 1/3 + c exp(-3t^2/2). 2. 2te^t - e^t. 3. rank=2, v_1 = (7,-10,0,1), v_2 = (-7,9,1,0). 4. unique for k not = -5, inf. many for k = -5, inconsistent for no values of k. 5. (1) LD; (2) LD; (3) LD; (4) LI. 6. (1,-1/7,5/7). 7. c_1 e^t (1,1) + c_2 e^{-t} (1,-1). 8. t = 25 (ln 3)/2, A_2 = 50 sqrt{3}/9.

o 08/05/04: Review: discussion of the sample final.

o 08/04/04: Applications: a coupled spring-mass system and mixing problems with two tanks. [Section 8.8] Answers: 2. x(t) = 3/10 sin 2t + 2/15 sin 3t, y(t) = 9/10 sin 2t - 4/15 sin 3t.

o 08/03/04: Nonhomogeneous linear systems of differential equations: method of variation of parameters. [Section 8.7] Answers: 2. x(t) = c_1 e^t (1,1) + c_2 exp(2t) (3,2) + ((3t-2)exp(2t)+3(t+1)e^t,2(t-1)exp(2t)+(3t+2)e^t). 6. x(t) = e^t (c_1(-1,1) + c_2(-1-2t,2t)+t(2t^2-3t-3,6t-2t^2)). 8. x(t) = c_1 (-2,1,0) + exp(3t)(c_2(1,0,2) + c_3(0,1,1) + (-(3t+2)/9,(33t+1)/9,3t)).

o 08/02/04: Discussion of Midterm IV. Solution by eigenvalues (the nondiagonalizable, repeated eigenvalues case). [Section 8.6] Answers: 2. x(t) = exp(-t)(c_1 (-1,1) + c_2 (1-2t,2t)). 8. x(t) = e^t ([0 0 1/2], [-1 -1-2t 2+4t+4t^2], [1 2t -3t-t^2]) (c_1, c_2, c_3). Here it is a 3x3 matrix, written row by row, times a column (c_1, c_2, c_3). 12. x(t) = c_1 exp(-5t) (36,-36,-5,6) + e^t(c_1 (0,0,1,0) + c_2 (0,0,t,-1) + c_3 (2,2,-t^2,2t+2)).

o 07/30/04: Midterm IV. Solution by eigenvalues (the diagonalizable complex case). [Section 8.5 (from the bottom p. 470)]

o 07/29/04: Doing homework problems from 8.4 and 8.5. Review: discussion of the sample test. [Sections 8.4 and 8.5 (through p. 470)]

o 07/28/04: General results on systems of linear differential equations: existence, uniqueness, the general solution, particular solution, IVP, etc. Solution by eigenvalues (the diagonalizable, real eigenvalues case). [Sections 8.4 and 8.5 (through p. 470)] Answers: 8.4.2. x(t) = ([exp(2t), (1+t) exp(2t)], [-exp(2t), -t exp(2t)]) (c_1, c_2). Here it is a 2x2 matrix, written row by row, times a column (c_1, c_2). 8.4.6. x_1(t) = (e^t cos 2t, e^t sin 2t) and x_2(t) = (e^t sin 2t, -e^t cos 2t). 8.5.6. Done in class. 8.5.16. x(t) = e^t (2,1) - exp(-5t) (-1,1). 8.5.20. (a) x_1' = x_2, x_2' = -cx_1 - bx_2. (b) p(lambda) = det (A - lambda I) = lambda^2 + b lambda + c.

o 07/27/04: Vector (matrix) formulation of first-order linear systems of differential equations. The new face of the Wronskian: linear independence of vector functions. [Section 8.3]

o 07/26/04: First-order linear systems of differential equations. Order reduction. Solving by elimination method. [Section 8.2] Answers: 10. x_1 = (1+2t) exp(3t) and x_2 = (3+2t) exp(3t). 12. x_1 = c_1 + c_2 exp(-t) + t(2-t)/2, x_2 = 2c_1 + c_2 exp(-t) + 1 -t^2. 14. x_1' = tx_2 + cost t, x_2' = x_3, x_3' = -x_1 + tx_2 + exp(t) cos t. 16. x_1' = x_2, x_2' = -x_1 -2tx_2 + cos t. 20. x_1 = cos t(3 sin t + 4), x_2 = sec t(sin^2 t - cos^3 t + sin 2t + 2t + 1).

o 07/23/04: Diagonalization. Diagonalizable = nondefective. Applying diagonalization to solving systems of differential equations. [Section 6.7] Answers: 2. Not diagonalizable. 4. S = ([i -i], [1 1]), diag(-4i,4i). 16. x_1 = c_1 exp(4t) - c_2 exp(8t), x_2 = c_1 exp(4t) + c_2 exp(8t). 20. x_1 = c_1 exp(-2t) + c_2 exp(3t), x_2 = c_1 exp(-2t) +c_3 exp(3t), x_3 = c_1 exp(-2t) - 4c_3 exp(3t).

o 07/22/04: More theory on eigenvalues and eigenvectors: defective and nondefective matrices, eigenspaces, algebraic and geometric multiplicities. [Section 6.6] Answers: 8. Eigenvalues: 3, 4i, -4i. The eigenspaces are all one-dimensional. Nondefective. 12. Eigenvalues: 0 and 2. The eigenspaces are all one-dimensional. Defective.

o 07/21/04: Eigenvalues and eigenvectors. What they are and how to find them. [Section 6.5] Answers: 4. 2 and 5. 6. For theta = 0 , lambda = 1, all nonzero vectors; and theta = pi, lambda = -1, all nonzero vectors. 12. lambda = 2 - 3i, v = {(t,it) : t nonzero complex number}; lambda = 2 + 3i, v = {(r,-ir): r nonzero complex number}. 14. 1, {(0,s,s) : s not 0}; 3, {(0,-t,t) : t not 0}. 16. -1, {(r,3r,4r): r not 0}; 1, {(3s,-5s,0): s not 0}; 3, {(-t,t,0): t not 0}.

o 07/20/04: Linear transformations in the plane R^2. Geometric interpretation. Transfers of regions. Elementary linear transformations. Elementary matrices. Factorization of a nonsingular matrix into the product of elementary ones. [Section 6.2] Answers: 8. T(x) = A x = M_1(-1)M_2(-1)x, a reflection in the x-axis, followed by a reflection in the y-axis. 10. T(x) = Ax = A_{12}(3)A_{21}(-1)M_2(-2), a reflection in the x-axis, followed by a stretch in the y-direction, followed by a shear parallel to the x-axis, followed by a shear parallel to the y-axis. 12. T(x) = Ax = A_{12}(1)M_1(-1)A_{21}(1)x, a shear parallel to the x-axis, followed by a reflection in the y-axis, followed by a shear parallel to the y-axis.

o 07/19/04: Mappings. Linear transformations: definition, properties, correspondence with matrices. [Section 6.1] Answers: 12. A = ([1 3], [2 -7], [1 0]), reading the matrix along the rows down from the top. 14. A = [1 5 -3]. 16. T(x_1, x_1, x_3) = (2x_1 + 2X_2 - 3x_3, 4x_1 - x_2 + 2x_3, 5x_1 + 7x_2 - 8x_3). 22. T(ax^2 + bx + c) = ax^2 + (-a+2b-2c)x + 3c.

o 07/16/04: Midterm Test III. Bases and dimension. [Section 5.6] Answers: 10. 2. 32. {exp(-3x), x exp(-3x)}. 34. The general vector in S: c(sin 4x + 5 cos 4x); a basis for the solution space: {sin 4x + 5 cos 4x, cos 4x}.

o 07/15/04: Linear independence in R^n (relation to homogeneous linear systems) and in function spaces (Wronskian). Review of Sample Midterm III. [Section 5.5 (starting from p. 306)]

o 07/14/04: The span of a set of vectors. Linear independence: definition and examples. [Sections 5.4 (from p. 296 and Problems 21, 25) and 5.5 (through p. 305)] Answers: 5.5.4. LD; v_1 - 2 v_2 + v_3 = 0. 5.5.8. LI. 5.5.24. {v_2, v_3, v_4}.

o 07/13/04: Subspaces, including the null space of a matrix and the general solution to a homogeneous linear second-order differential equation. Spanning (generation) and spanning sets. [Sections 5.3 (starting from Theorem 5.3.3 and Problem 5.3.21) and 5.4 (through p. 295 and Problem 17)]

o 07/12/04: Definition of a vector space. Subspaces. [Sections 5.2 and 5.3 (before Theorem 5.3.3 and Problem 5.3.21, which will be assigned on Tuesday)] Answers: 5.2.12. Hold: A1, A2, A5, A6. The other axioms fail.

o 07/09/04: The adjoint method. Cramer's rule. Vectors in R^n. [Sections 4.3 (starting from The Adjoint) and 5.1] Answers: 5.1.2. v = (22,-19,-53,39); -v = (-22,19,53,-39).

o 07/08/04: Further properties of determinants, elementary column operations. Minors, cofactors. Cofactor expansions. [Sections 4.2 (after Theoretical Results...) and 4.3 (before The Adjoint)] Answers: 4.3.16. 0. 4.3.22. (a) 11; (b) the 2x2 matrix with rows [5 -4] and [-1 3]; (c) rows [5 -1], [-4 3]; (d) 1/11 times the matrix from (c). 4.3.44. When none of a,b,c is 0, x_1 = (a-b+c)(a+b-c)/(4bc), x_2 = -(a-b-c)(a+b-c)/(4ac), x_3 = -(a-b-c)(a-b+c)/(4ab); when two of a,b,c are 0, but not the third one, no solutions; when a = 0, but b,c are not, {(0,1-r,r): r is any real number}; when only b = 0, and when only c = 0, the solution has a similar form.

o 07/07/04: Determinants and their row properties. Determinants and singular matrices. [Sections 4.1 and 4.2 (through Theoretical Results...)] Answers: 4.1.4. 0. 4.1.8. 48. 4.1.12. 42e^t. 4.2.6. -72. 4.2.14. 0; singular. 4.2.22. {-1,1,4}.

o 07/06/04: Midterm II handed out. The inverse of a matrix: definition and theory. Gauss-Jordan technique of finding the inverse. [Section 3.6] Answers: 8. The inverse is the matrix whose rows are [-43,-4,13], [-24,-2,7], and [10,1,-3]. 18. (4,-1). 28. Solution discussed in class.

o 07/02/04: Midterm II. Homogeneous linear systems. [Section 3.5 (last section therein)]

o 07/01/04: Gaussian and Gauss-Jordan elimination. Review and discussion of Sample Midterm II. [Section 3.5 (before Homogeneous Linear Systems)] Answers: 8. No solutions. 20. {(3+r-t,-r-1,r,t) : r and t real numbers}.

o 06/30/04: Elementary row operations and row-echelon matrices. The rank of a matrix. RREF. [Section 3.4]

o 06/29/04: Problem 4 from Midterm I. Matrices: basics, operations, relation to systems of linear equations [Sections 3.2-3.3]

o 06/28/04: Forced oscillations of a spring-mass system. Basics of matrices (assigned as home reading) [Sections 2.6 (forced oscillations) and 3.1]

o 06/25/04: Midterm exam I. Free oscillations of a spring-mass system. [Section 2.6 (free oscillations)]

o 06/24/04: A shortcut for when you have only one trigonometric function on the right-hand side, continued. Review of Chapters 1 and 2 (through 2.5) [Section 2.5]

o 06/23/04: Undetermined coefficients: the general complex and real cases; a shortcut for when you have only one trigonometric function on the right-hand side. [The rest of Section 2.4 and Section 2.5]

o 06/22/04: Undetermined coefficients: the case of no trigonometric functions. [Part of Section 2.4 not dealing with sines and cosines and complex roots of the auxiliary equation]

o 06/21/04: Second-order homogeneous linear DEs with constant coefficients. [Section 2.3]

o 06/18/04: Bernoulli equations. Second-order linear DEs: generalities. [Sections 1.8 (from Bernoulli) and 2.1]

o 06/17/04: Mixing problems. Electric circuits. First-order homogeneous DEs. [Sections 1.7 and 1.8 (before Bernoulli)]

o 06/16/04: Population growth: Malthusian and logistic. First-order linear DEs. [Sections 1.5 and 1.6]

o 06/15/04: The geometry of DEs: slope fields; existence and uniqueness. Separable DEs.[Sections 1.3 and 1.4]

o 06/14/04: How DEs arise: Newton's second law of motion, Hooke's law, orthogonal trajectories. Terminology. [Sections 1.1 and 1.2]


Last modified: Fri Aug 6 15:57:09 CDT 2004