General:
 Course info
General Syllabus
Daily Syllabus
Installing Matlab
Short Matlab tutorial
Homework:
HW 2 due Fri 1/29
HW 3 due Fri 2/5
HW 4 due Fri 2/12
HW 5 due Fri 2/19
HW 6 due Fri 2/26
HW 7 due Fri 3/4
HW 8 due Mon 3/28
HW 9 due Mon 4/4
Laplace Transforms
HW 10 due Fri 4/8
HW 11 due Fri 4/15
HW 12 due Fri 4/22
HW 13 due Wed 5/2
Figures for HW 13
Exam Reviews:
Review Exam 1
Review Exam 2
Review Exam 3
Review Final Exam
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MATH 316: Lecture Summaries and Homework
Lecture 1: Classification of Differential Equations. Course goals. Wed, Jan 20, 2016.
ODE vs PDE, linear vs nonlinear, order of a DE. Solutions.
Lecture Summary,
HW: S 1.3: 1-6,7,8,11,14,18,19. Due: Monday 1/25.
Lecture 2: Mathematical models. Fri Jan 22, 2016.
Mechanical systems, population models, mixing. Dimensions and units.
Lecture Summary,
HW: S 1.1: 21a,22,23,24a. Due: Monday 1/25.
Lecture 3: First order equations. Direction fields. Mon Jan 25, 2016.
First order ODEs. Direction fields. Autonomous equations. Equilibria.
Lecture Summary,
HW 2: 1-4 Due: Friday 1/29.
Lecture 4: Autonomous equations. Phase line. Wed Jan 27, 2016.
Use phase line to plot direction fields for 1st order autonomous ODEs.
Find equilibria (in autonomous systems) and their stability.
Lecture Summary,
HW 2: 5-8 Due: Friday 1/29.
Lecture 5: Separation of variables. Fri Jan 29, 2016.
Using the Chain Rule. Division by zero. Solving the logistic equation.
Lecture Summary,
HW 3: 1-4 Due: Friday 2/5.
Lecture 6: Linear equations - method of integrating factors. Mon Feb 1, 2016.
Main idea of method. Examples. Limiting behaviours.
Lecture Summary,
HW 3: 5-9 Due: Friday 2/5.
Lecture 7: Modeling. Linear vs nonlinear equations. Wed Feb 3, 2016.
Using phaseline and solution methods to explore mathematical models.
HW 3: 10-12 Due: Friday 2/5.
Lecture 8: Eulers method. Fri Feb 5, 2016.
Derivation. Example by hand. Matlab function.
Lecture Summary,
HW 4: 4. Due: Friday 2/12.
Lecture 9: Modified Euler method. Mon Feb 8, 2016.
Derivation. Matlab function. Matlab results.
Lecture Summary,
Lecture 10: Review. Wed Feb 10, 2016.
Lecture 11: Exam 1. Fri Feb 12, 2016.
Lecture 12: 2nd order linear, constant coefficient, homogeneous. Mon Feb 15, 2016.
Constant coefficient, homogeneous. Characteristic equation.
General solution. Example. Linear operators.
Case 1: real, distinct roots
Lecture Summary,
HW 5: 1-5. Due: Friday 2/19.
Lecture 13: The Wronskian. Wed Feb 17, 2016.
Constant coefficient, homogeneous. Characteristic equation.
The Wronskian and fundamental sets of solutions.
Lecture Summary,
HW 5: 6-8. Due: Friday 2/19.
Lecture 14: 2nd order linear, constant coeffiecient, homogeneous. Fri Feb 19, 2016.
Case 2: complex roots. Review of complex numbers. General real solution.
Lecture Summary,
HW 6: 1-3. Due: Friday 2/26.
Lecture 15: 2nd order linear, constant coeffiecient, homogeneous. Mon Feb 22, 2016.
Finding amplitude and phaseshift of periodic functions.
Case 3: repeated roots. Summary of all cases.
Lecture Summary,
HW 6: 4-5. Due: Friday 2/26.
Lecture 16: 2nd order linear homogeneous - reduction of order. Wed Feb 24, 2016.
Given one solution to a 2nd order linear homogeneous differential equation,
this method yields a second solution that completes the set of solutions
to give a general solution.
Lecture Summary,
HW 6: 6-7. Due: Friday 2/26.
Lecture 17: 2nd order linear nonhomogeneous - undetermined coefficients.
Fri Feb 26, 2016.
Lecture Summary (17+18)
Lecture 18: 2nd order linear nonhomogeneous - undetermined coefficients.
Mon Feb 29, 2016.
What to do when the guess y_p is part of the homogenous solution y_H.
Resonance.
HW 7: 1. Due: Friday 3/4.
Lecture 19: 2nd order linear nonhomogeneous - variation of parameters.
Wed Mar 2, 2016.
Method of variation of parameters. Examples.
Lecture Summary,
HW 7: 2. Due: Friday 3/4.
Lecture 20: Linear Mechanical Oscilators: springs - damped case.
Fri Mar 4, 2016.
Damped springs, no forcing: overdamped, underdamped and critically damped.
Damped springs, forced: transient and forced parts of solution. Large amplitude for
small damping and forcing frequency close to spring frequency.
Lecture Summary (20+21)
Lecture 21: Linear Mechanical Oscilators: springs - undamped case.
Mon Mar 7, 2016.
Undamped springs witout forcing: oscillations.
Undamped springs with forcing: Beats. Resonance.
Lecture 22: Review. Wed Mar 09, 2016.
Lecture 23: Exam 2. Fri Mar 11, 2016.
SPRING BREAK
Lecture 24: Laplace Tranform - definition, linearity, applications.
Mon Mar 21, 2016.
Definition and examples. Linearity. Solving an initial value problem.
Lecture Summary
Lecture 25: Laplace Tranform - building and using table of transforms.
Wed Mar 23, 2016.
More basic transforms. Convergence. Solving another initial value problem.
Lecture Summary
Lecture 26: Laplace Tranform - the shift formula. functions defined piecewise.
Fri Mar 25, 2016.
L[exp(ct)g(t)]. Example. Piecewise functions and u_c(t).
Lecture Summary
Lecture 27: Laplace Tranform - discontinous forcing.
Mon Mar 28, 2016.
L[u_c(t)g(t-c)]. Example.
Lecture Summary,
Example
Lecture 28: Laplace Tranform - impulse functions.
Friday April 1, 2016.
L[delta]=1. Example.
Lecture Summary
Lecture 29: Laplace Tranform - convolutions.
Friday April 1, 2016.
L[f*g]=L[f]L[g]. Example.
Lecture Summary
Lecture 30: Linear Algebra - introduction.
Monday April 4, 2016.
Matrices, operations, 2x2 linear systems.
Lecture Summary
Lecture 31: Solving x'=Ax. The eigenvalue problem.
Monday April 4, 2016.
Write a second order equation in matrix form. Look for
solutions x=e^{\lambda t}v => Eigenvalue problem.
Lecture Summary
Lecture 32: The phase plane. Example.
Lecture Summary
Lecture 33: Real distinct eigenvalues. Equilibria, det(A), eigenvals.
Lecture Summary
Lecture 34: Real distinct eigenvalues -- saddles, nodes, line of equilibria.
Lecture Summary
Lecture 35: Complex eigenvalues -- stable and unstable spirals, centers.
Lecture Summary
Lecture 36: Repeated eigenvalues -- degenerate nodes.
Lecture Summary
Lecture 38: Review. Fri Mar 09, 2016.
Lecture 39: Exam 3. Mon Mar 11, 2016.
Lecture 40: Nonlinear autonomous systems - linearization.
Lecture Summary
Lecture 41: Nonlinear autonomous systems - Examples. Competing species.
Lecture Summary
Lecture 42: Nonlinear autonomous systems - Predator-prey. Pendulum.
Lecture Summary
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