Math 2171 (Ordinary Differential Equations) Section 004 - Fall 2009

MATH 2171 (Ordinary Differential Equations) Section 004 - Fall 2009 - UNC Charlotte

Class meets: Mondays and Wednesdays 12:30pm-1:45pm, Duke 259.
My Email:

My Office: Fretwell 370E. Enter main doors to the math department on the third floor of Fretwell, then turn left by the front desk. My office is down the hallway on the left. Refer to the picture on the right.
My Office Phone: 704-687-3874
My Office Hours: Tuesdays 12:30pm-2:30pm, Thursdays 2:30pm-3:30pm, and by appointment.

Tutoring / Help

Please do not hesitate to stop by my office hours or email me for help. However, if I'm not available, you can always get FREE tutoring from the Math Learning Center (MLC). Its hours of operation this semester are:

Monday: 11:00am-2:00pm in Fretwell 315.
Monday: 5:00pm-7:00pm in Fretwell 307.
Tuesday-Thursday: 11:00am-2:00pm and 5:00pm-7:00pm in Fretwell 315
Friday: 12:00pm-2:00pm in Fretwell 315.

If you cannot get help from me or the MLC, you can always use Tutorial Services on campus. It is also a free service, and you get one-on-one tutoring for an hour at a time.

Exams

Exam 1 Study Guide,   Exam 1,   Exam 1 - KEY,   Exam 1 - Extra Credit,   Exam 1 - Extra Credit - KEY
Exam 2 Study Guide,   Exam 2,   Exam 2 - KEY,   Exam 2 - Extra Credit,   Exam 2 - Extra Credit - KEY
Final Exam Study Guide,   Final Exam,   Final Exam - KEY,   Final Exam - Extra Credit,   Final Exam - Extra Credit - KEY

Lab

Here's the lab and its associated MATLAB .m file. The lab is due in-class on the last class day, Wednesday 12/09/2009.

Lectures

Lecture 1: Classification of differential equations, initial conditions and particular solutions, radioactive decay. Direction fields, integral curves, autonomous DEs, the method of isoclines.

Lecture 2: Worked examples of first order DEs. Numerical approximation with Euler's method, limitations of Euler's method. Classification of DEs.

Lecture 3: Example of Euler's method. Second order linear equations with constant coefficients. First order linear equations and integrating factors.

Lecture 4: Examples of first order linear equations. Variation of parameters. Separable equations. Interpretation of differential symbols in separable equations. Explicit vs. implicit solutions. Examples.

Lecture 5: Examples of separable equations. Using initial conditions to choose the branch of the square root function. Modeling with first order equations. Input/output models. The Malthusian model for population growth. Applications to finance and chemical mixture problems. Qualitative and physical interpretation of the results.

Lecture 6: Modeling examples. Differences between linear and nonlinear equations. Blowing up to infinity in finite time. Existence and uniqueness for first order linear DEs.

Lecture 7: Existence and uniqueness examples. Autonomous equations and population dynamics. Critical points. Determining the qualitative behavior of the solution to an autonomous DE without solving it. Stable, asymptotically stable, semistable, and unstable solutions. Carrying capacity. Examples.

Lecture 8: Examples of autonomous equations. The total differential and exact differential equations. Solution method for exact equations. Necessary conditions for a DE to be exact. Examples.

Lecture 9: Exact equations (again) with examples.

Lecture 10: Examples of exact equations. Review for Exam 1.

Lecture 11: Vectors, matrices, and scalars. Dot products. Matrix-vector multiplication. Examples.

Lecture 12: Matrix and vector examples. Solving a system of 2 equations in 2 unknowns via Gaussian elimination. Geometric interpretation of systems of equations. The identity matrix. Homogeneous vs. nonhomogeneous systems of equations. Conditions for existence and uniqueness of solutions to systems. Matrix inverses.

Lecture 13: Review of matrix algebra so far. Eigenvalues and eigenvectors. The characteristic equation for eigenvalues. Examples. Eigenvalues and eigenvectors of real matrices come in complex-conjugate pairs. Geometric vs. algebraic multiplicity of eigenvalues.

Lecture 14: Examples of eigenvalues and eigenvectors. Expressing a system of 2 first order linear DEs in matrix format. Homogeneous vs. nonhomogeneous systems, autonomous systems, critical points, coupled vs. decoupled systems. Transforming higher order equations into a system of first order equations. Solving a system of two first order linear DEs.

Lecture 15: Examples of transforming higher order DEs to systems. Examples of rewriting systems in matrix format. Solving systems of two first order linear DEs by using eigenvalues and eigenvectors. Principle of superposition. Wronskians and fundamental solution sets. Phase portraits. Stable nodes.

Lecture 16: Examples of systems. Saddles and stable nodes. Component plots. Euler's formula and complex eigenvalues and eigenvectors. Stable and unstable spirals.

Lecture 17: Examples of systems. Second order equations with constant coefficients. The characteristic equation and three cases: (1) real and distinct roots, (2) real and repeated roots, (3) complex conjugate roots. Solution method for cases (1) and (2). Classification of second order DEs.

Lecture 18: Examples of classification, existence and uniqueness theorems, Wronskians, and fundamental solution sets. Unstable nodes and a line of nodes. Handling case (3), i.e. complex conjugate roots. Cauchy-Euler equations.

Lecture 19: Examples of DEs with complex eigenvalues. Sprial point or sink. Examples of second order constant-coefficient and Cauchy-Euler equations.

Lecture 20: More examples of Cauchy-Euler equations. Mechanical and electrical vibrations. Amplitude, angular frequency, phase, period, and phase-shifts. Adding sines and cosines with the same phase but different amplitudes. Modeling mass-spring-dashpot systems with second order DEs. Three physical cases: (1) overdamped, (2) criticall damped, (3) underdamped. Solutions for each case.

Lecture 21: Solutions to mechanical systems problems. Nonhomgeneous equations and the method of undetermined coefficients. Linear operators, associated homogeneous equations, general vs. particular solutions.

Lecture 22: Examples of nonhomogeneous systems and undetermined coefficients. Kernels and the Laplace transform. Integrating complex-valued functions. Calculating the Laplace transform. Linearity of the Laplace transform. Piecewise continuous functions. Functions of exponential order and existence of the Laplace transform. Examples.

Lecture 23: Examples of Laplace transforms. Various properties of the Laplace transform, including translation in space (shifting property), transforms of derivatives, and the transform of t^n*f(t).

Lecture 24: Examples of Laplace transform properties. Transforms of integrals. Taking the Laplace transform of an initial value problem. Inverse Laplace transforms. Linearity of the inverse transform. Partial fractions.

Lecture 25: Solving initial value problems with the Laplace transform. Partial fractions examples. Convolution integrals. Properties of the convolution. The Convolution Theorem. Solving integro-differential equations using the Convolution Theorem and Laplace transforms.

Lecture 26: Examples of solving DEs with Laplace transforms. Examples of convolution integrals and integro-differential equations. The Laplace transform of discontinuous functions and periodic functions. Indicator functions and the Heaviside step function. Representing piecewise-defined functions as combinations of indicator functions. Translation in time for Laplace transforms.

Lecture 27: Examples of transforms of discontinuous and periodic functions. Using the Laplace transform to solve DEs with discontinuous right-hand-sides. Impulse functions (e.g. the Dirac delta function). Sifting property of the delta function. Solving IVPs with a delta-function right-hand-side.

Graded Homework

Graded Homework #1 -- DUE Wednesday, 09/02/2009 in class.
Graded Homework #1 -- KEY
Graded Homework #2 -- DUE Wednesday, 09/23/2009 in class.
Graded Homework #2 -- KEY
Graded Homework #3 -- DUE Wednesday, 10/28/2009 in class.
Graded Homework #3 -- KEY
Graded Homework #4 -- DUE Wednesday, 12/09/2009 in class.
Graded Homework #4 -- KEY

Other Handouts

Daily homework #20 (For solutions to all problems, see Lecture 21)

Daily Homework

In the following daily homework assignments, a star (*) indicates an optional/bonus problem.

Daily homework #1: 1.1/1-11 odd, 15-19 odd, 22, 23, 25, 27*-31* with technology; 1.2/1-4, 11, 13, 14, 16-20.
Daily homework #2: 1.3/1-15 odd; 1.4/1-11 odd, 14*, 15-19 odd (Hint for #14: you need to use the Fundamental Theorem of Calculus and the product rule to differentiate y).
Daily homework #3: 2.1/1-31 odd, 39*-43*
Daily homework #4: 2.2/1-23 odd, 30*, 31*-37* odd
Daily homework #5: 2.3/1-5 odd, 6*, 7-9, 11*, 12*, 16, 23*, 25*, 26*, 32**
Daily homework #6: 2.4/1-15 odd, 23, 27*-31* odd, 32, 33, 34**
Daily homework #7: 2.5/1-5 odd, 7*, 9-13 odd, 14, 15*, 23**, 26*
Daily homework #8: 2.6/1-15 odd (ignore part (c)), 17*, 18*
Daily homework #9: Finish daily homework #8 if you haven't already.
Daily homework #10: NONE. STUDY FOR EXAM #1.
Daily homework #11: Read sections 3.1 and 3.2.
Daily homework #12: 3.1/1-11 odd. Finish reading 3.1 and 3.2.
Daily homework #13: 3.1/13-35 odd, 37*.
Daily homework #14: 3.2/1-7 odd, 9-13 odd part (a) only, 15-18. Also, (re)read sections 3.2-3.3, and also 3.4 if you have time.
Daily homework #15: 3.3/1-23 odd, 25*. For help drawing phase portraits and component plots, see Examples 3-5 in setion 3.3. Read Appendix B and section 3.4.
Daily homework #16: 3.4/1-13 odd.
Daily homework #17: 4.1/1-5. Determine linearity, homogeneity, and autonomousness; 4.2/1-13 odd, 15*, 21, 23; 4.3/1-29 odd, 30, 31, 32*, 33*; Read section 4.4.
Daily homework #18: 4.4/11-25 odd, 27*, 28*.
Daily homework #19: 4.4/35-41 odd.
Daily homework #20: CLICK HERE.
Daily homework #21: Read section 4.6. Do 4.6/1-17 odd.
Daily homework #22: 5.1/6-12 even, 13-27 odd.
Daily homework #23: 5.2/1-9 odd, 11*, 13-21 odd, 22*.
Daily homework #24: 5.3/1-23 odd.
Daily homework #25: 5.4/1-9 odd, 11*, 13*, 15* (read pp.337-338); 5.8/1-11 odd, 13*. Read ahead sections 5.5, 5.6, and 5.7 for next week.
Daily homework #26: 5.5/1-23 odd.
Daily homework #27: 5.6/1-11 odd, 13*; 5.7/1-11 odd, 15*.