Differential Equations
This course is taught at the United States Naval Academy, Annapolis, MD.
I strongly recommend that you form study groups with other midshipmen in preparation for your Final Exam.
Wed 7 Dec (at 2359) is the last day that I am accepting LATE WebAssign assignments. You can continue to work on WebAssign assignments after that day for zero credit.
Inside the Navy's indoor ocean.
Practice exams were written by other faculty members, and per their wishes, I am not going to post the solutions to every practice exam.
You need to go to the Math Lab or MGSP for help, or work through the problems with other midshipmen and compare your techniques and your solutions. Practicing problems from scratch is the best way to really learn how to master the lesson objectives.
Furthermore, there are plenty of practice exams with complete solutions written by the author of the practice exam, so if you want, go through those practice exams first.
Example of an electrical circuit.
Welcome to SM212 Differential Equations:
Section 3021 MTWF (0955-1045), CH222,
Section 4021 MTWF (1055-1145), CH222,
Section 5021 MTWF (1330-1420), CH222.
We will be using Differential Equations with Boundary Value Problems (9th edition) by Dennis G. Zill. Since this course is for engineering, mathematics, and physics majors, the primary applications we will be focusing on are mass-spring systems and electrical circuits.
Professor Mee Seong Im
Office: Chauvenet Hall 342
Office Phone Number: (410) 293-6776
Email: im [at] usna [dot] edu
Walk-in weekly Extra Instruction (EI) in my office: Feel free to drop by and get extra assistance as needed.
Mondays: 1220 - 1320 (available with advanced notice; you may let me know before or after class as well)
Tuesdays: 1220 - 1320
Wednesdays: 1220 - 1320 (available with advanced notice; you may let me know before or after class as well)
Fridays: 1220 - 1320
I was told that the after-lunch time windows (1250-1320) on Mondays and Wednesdays are reserved for CO/SEL training time. Academics-related activities should be on Tuesdays or Thursdays (and presumably Fridays) as per the Provost.
I will however be available on Mondays and Wednesdays during lunch (1220-1320) in case you need extra help and would like to drop by. Please email me to let me know in advance.
Other Extra Instructions (EI): Request for them at least 2 business days in advance (this excludes evenings, weekends, and Federal Holidays). In your email, please give me at least 4 time-slots and days that work for you.
For all EIs, please attend prepared, with homework problems attempted in advance and with specific questions in mind.
Midshipmen Group Study Program (MGSP; free peer-tutoring and assistance from two amazing leaders), in CH155.
Tuesdays, 2000-2100, with MIDN Karl Florida
Thursdays, 2100-2200, with MIDN Maximus Lear
Math Lab, in CH130. This is run by the Department of Mathematics faculty members.
The Math Lab is open Mondays through Fridays, 1st through 6th period: 0755-1520. Walk-ins are welcome!
Academic Center, US Naval Academy, Annapolis, MD. Faculty and Staff (Professional Tutors).
The Academic Center offers free one-on-one professional tutoring during the day (appointments may be necessary).
Free evening professional tutoring is available (walk-ins are encouraged).
The Academic Center also offers tutoring via online platforms, like Google Meet and Zoom, if you cannot leave your room due to covid or some other reason.
Study Groups.
There are about 132 midshipmen taking this course. You are strongly encouraged to form study groups with other midshipmen.
Section Leaders:
Section 3021 MTWF, CH222
Section Leader: MIDN Yohan Afewerki
Assistant Section Leader: MIDN Kathryn (Katie) Jackowski
Assistant Section Leader: MIDN Tyler Sweatt
Section 4021 MTWF, CH222
Section Leader: MIDN Athena Dinh
Assistant Section Leader: MIDN Kevyn (Kai) Sung
Assistant Section Leader: MIDN Abigail (Abby) Palaia
Section 5021 MTWF, CH222
Section Leader: MIDN Jadiel Arana
Assistant Section Leader: MIDN Jackson Suddath
Assistant Section Leader: MIDN Nicholas (Kai) Widick
For those who registered for WebAssign incorrectly (that is, Cengage is asking you to pay), you have been dropped from that section. Now, please follow the instructions below.
How to Register for WebAssign through Blackboard (follow the instructions here): https://startstrong.cengage.com/webassign-blackboard-ia-no/
Once you are registered with your usna.edu email address, you should see that you have access to the homework assignments and the online textbook until the end of December 2022.
Note that you must purchase Access Code from USNA's bookstore or from Cengage.com since your trial period will end on 8/31/2022.
0.3306934 lb mass oscillating in air.
0.3306934 lb mass oscillating in water (it has lower frequency than in air due to the damping force (drag)).
A further discussion of pure resonance: an undamped oscillator (spring/mass problem) so β = 0.
Below each lesson, I will summarize the topics covered during each day.
Lesson 1 (Mon 22 Aug): Introduction to differential equations (Section 1.1)
Two types of differential equations: ODE has 1 or more unknown functions with respect to a single independent variable and PDE has 1 or more unknown functions with respect to 2 or more independent variables.
Leibniz notation, prime notation, dot notation, subscript notation.
The order of a differential equation (it's the order of the highest derivative in the equation).
Differential form (example: M(x,y) dx + N(x,y) dy = 0).
Lesson 2 (Tues 23 Aug): Intro to differential equations (Section 1.1)
n-th order differential equation in one variable F(x, y, y', y'',...,y^(n)) = 0 is a real-valued function of n+2 variables.
How to rewrite n-th order ordinary differential equation in the normal form.
n-th order linear and nonlinear differential equations in one variable.
A solution of an ODE is defined over an interval of definition, interval of existence, interval of validity, domain of the solution, commonly denoted by I.
Verify that the given function is a solution to the differential equation.
A function has the dual notion as a function and as a solution to a differential equation.
Sketching solution curves.
Lesson 3 (Wed 24 Aug): Intro to differential equations (Section 1.1), Initial value problems (Section 1.2)
Quiz 01
Implicit and explicit solutions.
Verify that the given 1-parameter family is an explicit solution to a linear 1st order differential equation.
Verify that the given 2-parameter family is an explicit solution to a linear second order differential equation.
Piecewise solutions of a linear first-order equation (which cannot be obtained by a single choice of one parameter). So they form another (new) family of solutions to a differential equation.
Solve linear first order equation subject to an initial condition y(x_0) = y_0.
Given a solution with 2-parameters for a linear second-order differential equation with two initial conditions, find what the two parameters must be equal to.
Your WebAssign grades were synchronized to Blackboard today. But as long you continue to work on the homework assignment, your old grade will automatically be replaced with the new grade, and will be updated in Blackboard (twice a day). So please do not worry so much about your homework grade now.
Your Quiz 01 has been graded and your grades have been entered in Blackboard. Your and your partner's names have been noted on the Google Spreadsheet since I am keeping track of this information (remember to be partners with a different midshipman for every quiz or you will lose points).
Announcement (Thurs 25 Aug): SM212 MGSP schedule has been posted.
It will begin on Sunday 28 Aug 2022. Scroll up to see the schedule. I encourage you to attend with your friends, and bring questions.
Lesson 4 (Fri 26 Aug): Separable equations (Section 2.2) and First order linear differential equations (Section 2.3)
Solve first-order separable equations (both with initial value and without initial value conditions).
Discuss first-order linear differential equation (DE).
Definition of standard form for linear first-order DE: dy/dx + P(x)y = f(x).
Discuss the technique to solve non-separable linear first-order DE.
First, put the linear first-order DE in standard form dy/dx + P(x)y = f(x).
Compute the integrating factor: μ(x) = e^{∫P(x)dx}.
Multiply μ(x) across each term of the DE in the standard form.
The LHS becomes d/dx(μ(x) y(x)) while the RHS becomes μ(x)f(x).
Integrate d/dx(μ(x) y(x)) = μ(x)f(x) with respect to x.
Solve for y(x).
Lesson 5 (Mon 29 Aug): Direction fields (Section 2.1.1) and Euler's method (Section 2.6)
Do one more problem from Section 2.3: how to solve non-separable linear first-order DE.
How to detect a solution curve when given a direction field to a first-order DE and an initial condition.
Given a first-order DE and an initial-value y(x_0)=y_0, use linearization to approximate the y-value to a given point.
Lesson 6 (Tues 30 Aug): Applications of linear models: exponential growth and decay and cooling (Section 3.1)
Bacteria growth: a culture has an initial amount P(0) = P_0 and 1 hour later, it has P(1 hour) = 3/2 P_0. The rate of growth is proportional to the number of bacteria P(t) present at time t, i.e., dP/dt = kP, where k is the constant of proportionality. When does the number of bacteria triple?
Half-life of plutonium example.
The cooling of a cake from the oven to the room temperature.
I have decided to give you more time and change the due date for the WebAssign assignment "2.3 First-Order Equations: Linear" from 8/30/22 to 8/31/22.
Lesson 7 (Wed 31 Aug): Applications of linear models: Mixing (Section 3.1)
Quiz 02
The mixture of two salt solutions. What happens to the amount of salt in the tank in the long run?
If the brine solution is being pumped out at a slower rate, what happens to the amount of salt in the tank in the long run? What can you say about the mixing tank with the capacity to store at least 300 gallons of brine solution? [Answer: If there is an increasing amount of salt saturation in the salt tank in the long run, then some parts of the tank probably need cleaning or replacement due to a buildup of residues, minerals, or rust/metallic deposits. So this is a great way to detect malfunctioning equipment.]
Your Quiz 02 has been graded and your grades have been entered in Blackboard. Your and your partner's names have been noted on the Google Spreadsheet since I am keeping track of this information (remember to be partners with a different midshipman for every quiz or you will lose points).
Lesson 8 (Fri 2 Sept): Applications of linear models (Section 3.1)
Another mixing problem example where the amount of fluid entering the tank does not equal the amount of fluid leaving the tank. I will set-up the problem and a midshipman will solve the rest of the problem at the board.
Another midshipman leads the class on WebAssign "3.1 Applications: Growth and Decay, Cooling" Problem #5.
I've given you an extension on WebAssign "1.3 Mathematical Models" and "3.1 Applications: Growth and Decay, Cooling". They are not due tonight; they are due on Tuesday at 2359.
Lesson 9 (Tues 6 Sept): Homogeneous linear equations (Section 4.1.2)
Homogeneous versus inhomogeneous n-th order linear differential equation.
Differential operator: D = d/dx.
The general n-th order differential operator L is a linear operator.
Theorem: superposition principle for homogeneous equations.
Fact: if y(x) is a solution of L(y) = 0, then c y(x) is also a solution of L(y) = 0.
Fact: given L(y) = 0, the trivial solution y(x) = 0 is always a solution of L(y) = 0.
A set of functions being linearly dependent on an interval I.
If a set of functions is not linearly dependent on an interval I, then they are linearly independent.
Lesson 10 (Wed 7 Sept): Homogeneous linear equations (continued) (Section 4.1.2)
Quiz 03
4th period only: CDR Jennifer Matthews (The Chair of the Mathematics Department) will visit our class.
Show that the general 2nd-order differential operator L is linear.
Do another example to show whether or not a set of functions is linearly dependent or linearly independent on its interval I of definition.
The definition of Wronskian and compute some examples.
Due to a lot of work, I will get Quiz 3 graded by tomorrow.
Announcement (Thurs 8 Sept):
Quiz 3 will be graded tonight.
Lesson 11 (Fri 9 Sept): Linear homogeneous differential equations: real roots (Section 4.3)
Quiz 3 has been graded this morning.
Discuss when to use Wronskian to determine whether or not a set of functions is linearly independent or linearly dependent on an interval I.
Discuss how to find the general solution to 1st-order homogeneous linear DE with constant coefficients. Solutions are of the form y(x) = e^{mx}.
Discuss how to find the general solution to 2nd-order homogeneous linear DE with constant coefficients. Solutions are of the form y(x) = e^{mx}.
Case 1: the radical in the quadratic formula for m is positive (two distinct real roots).
Case 2: the radical in the quadratic formula for m equals zero (repeated real roots).
Do plenty of examples.
Lesson 12 (Mon 12 Sept): Linear homogeneous differential equations: complex roots (Section 4.3)
Discuss how to find the general solution to 2nd-order homogeneous linear DE with constant coefficients. Solutions are of the form y(x) = e^{mx}.
Case 3: the radical in the quadratic formula for m is negative (complex conjugate roots).
Do plenty of examples.
Lesson 13 (Tues 13 Sept): Mass-spring systems (free undamped motion) (Section 5.1.1) Review Linear homogeneous differential equations: complex roots (Section 4.3)
Discuss how to find the general solution to 3rd-order homogeneous linear DE with constant coefficients.
Discuss how to find the general solution to 4th-order homogeneous linear DE with constant coefficients.
Lesson 14 (Wed 14 Sept): Mass-spring systems (free undamped motion; continued) (Section 5.1.1)
Quiz 04
I'm participating in the Mid-for-a-Day Program, following MIDN Kai Widick all day. No EI today.
Hooke's Law: F = -k s, k > 0 is the spring constant and s = amount of elongation with some mass attached to a spring and in equilibrium position. So |F| = k|s|.
W (weight in lbs) = m g, where g = 9.8 m/s^2 or g = 32 ft/s^2.
Weight balanced by restoring force: m g = k s.
Newton's Second Law of Motion: the net (resultant force) on a moving body of mass m is the sum of all forces acting on m: m d^2x/dt^2 = -k(x+s) + mg, or m d^2x/dt^2 = - k x.
To find the equation of motion, solve for m d^2x/dt^2 + k x = 0, or d^2x/dt^2 + (k/m) x = 0.
Note that ω^2 = k/m, so the period of motion is T = 2π/ω (in sec). This is the time it takes the mass to execute one complete cycle of motion.
The frequency is f = 1/T = ω/2π. This is the number of cycles completed in 1 second.
Lesson 15 (Fri 16 Sept): Mass-spring systems (free undamped motion) (Section 5.1.1) and Mass-spring systems (free damped motion) (Section 5.1.2)
Review 5.1.1. and do a problem.
The damping force is proportional to a power of the instantaneous velocity, i.e., β dx/dt, where the damping constant β > 0.
So a differential equation corresponding to the equation of motion is m d^2x/dt^2 + β dx/dt + k x = 0.
Lesson 16 (Mon 19 Sept): Review
When you have x(t) = C_1 cos(ωt) + C_2 sin(ωt), with C_1 not necessarily equal to C_2, then you can write x(t) compactly as x(t) = A sin(ωt + ϕ), where the amplitude of x(t) is A = √ (C_1^2 + C_2^2) and see above for ϕ.
Extreme displacement of a mass in an oscillation problem is the greatest distance away from the equilibrium position. To find the extreme displacement (above or below the equilibrium position), find time t when x'(t) = 0.
Extra MGSP will be held by MIDN Karl Florida tonight at 2100-2200 in CH120.
Lesson 17 (Tues 20 Sept): Review
Consider m x'' + β x' + k x = 0. Divide through by m to get x'' + (β/m) x' + (k/m) x = 0. Let 2λ = β/m and ω^2 = k/m. Since x'' + 2λ x' + ω^2 x = 0, m = - λ ± √(λ^2 - ω^2).
If λ^2 - ω^2 > 0, then the spring system is overdamped (smooth, nonoscillatory motion because the damping constant is much bigger than the spring constant).
If λ^2 - ω^2 = 0, then the spring system is critically damped (if you decrease the damping constant β, then x(t) would result in oscillatory motion).
If λ^2 - ω^2 < 0, then the spring system is underdamped (x(t) is oscillatory, but because of e^{-λt}, where λ >0, lim_{t → ∞} x(t) = 0. That is, the amplitudes of x(t) decrease as t → ∞).
MGSP will be held by MIDN Karl Florida tonight at 2000-2100 in CH155.
Lesson 18 (Wed 21 Sept): Test 1
Only a USNA-issued calculator TI-36X Pro is allowed.
There are 8 problems on your Test 1. Some problems have multiple parts.
There are multiple versions of the test-- some are 6 pages while others are 7 pages.
For some problems, you need to show work to get the full credit (it says on there explicitly if you need to show work). If you don't show work, you will lose a large proportion of the points.
The best way to study is to do as many problems as possible, not stare at a problem and its solution.
If you write both wrong solutions (incorrect work) and correct solutions on your test, then you will lose points for writing wrong solutions.
Both versions of Test 1 has been posted in my Google Drive.
Announcement (Thurs 22 Sept):
Your exams are currently being graded. I am aiming to get them done by today.
Lesson 19 (Fri 23 Sept): Nonhomogeneous linear equations (Section 4.1.3)
Particular solution to Ly = g(x) and complementary solution (fundamental set of solutions) to Ly = 0, where L is an nth order linear differential operator.
Let y_p be a particular solution of Ly = g(x) and let y_1, y_2, ..., y_n be a fundamental set of solutions of Ly = 0. Then the general solution to Ly = g(x) is y(x) = C_1 y_1 + C_2 y_2 + ... + C_n y_n + y_p.
Superposition principle for nonhomogeneous linear equations.
Generalized superposition principle for nonhomogeneous linear equations.
I am not done grading your Test 1. I will grade your tests into this weekend.
Announcement (Sat 24 Sept):
On your test: grading is still in-progress.
Announcement (Sun 25 Sept):
Your Test 1 has been graded. It will be passed back tomorrow.
Lesson 20 (Mon 26 Sept): Undetermined coefficients (Section 4.4)
Let's try again. 4th period only: CDR Jennifer Matthews (The Chair of the Mathematics Department) will visit our class.
Test 1 average: 80.39919, standard deviation: 14.4325.
1 point bonus was added to your test grade since the highest grade is 99 out of 100 points. Test 1 average after the bonus: 81.39919, standard deviation: 14.4325.
Discuss why Wronskian doesn't always work to determine linear dependence when given a random set of functions.
Find a particular solution to nth order nonhomogeneous linear differential equation.
Find the general solution to nth order nonhomogeneous linear differential equation.
Go to search box on the bottom left of your laptop, type Software Center, and install Mathematica. We will begin to work with this tomorrow, Tuesday 27 Sept 2022.
The cutoff date for your 6-week grades is today. Your 6-week grades were submitted today, after I have completed grading your exams.
Lesson 21 (Tues 27 Sept): Undetermined coefficients (Section 4.4)
Do more examples.
Check our work using Mathematica.
Lesson 22 (Wed 28 Sept): Undetermined coefficients (Section 4.4)
Quiz 05
More examples.
Your Quiz 5 has been graded.
Lesson 23 (Fri 30 Sept): Mass-spring system: with external force, resonance (Section 5.1.3)
Examples and Mathematica demonstrations
Lesson 24 (Mon 3 Oct): Series electrical circuits (Section 5.1.4)
i = i(t) = current in the LRC-series electrical circuit (ampere or A)
q = q(t) = charge (coulombs or C)
i = dq/dt
E(t) = impressed voltage (volts or V)
Inductors (henries or h), passive component in electrical circuits to store energy in the form of magnetic energy when electric currents flow through it (coil/helix) with 2 terminals, used to increase magnetic flux through the circuit, turn off current and the magnetic field starts to collapse and magnetic energy converts to electrical energy, resists sudden change in current, smooths out the current drop in a circuit:
L di/dt
Resistors (ohms or Ω), 2-terminal electrical component providing electrical resistance:
R i
Capacitors (farads or f), store energy (all the way up to the voltage amount in the battery in the circuit) in the form of static charge and resist sudden change in voltage:
(1/C) q
Kirchhoff's Second Law: the total voltage impressed on a circuit/closed loop is equal to the sum of these voltage drops in the loop.
So L i'(t) + R i(t) + 1/C q(t) = E(t).
Kirchhoff's Second Law ( since i(t) = q'(t) ): L q''(t) + R q'(t) + 1/C q = E(t).
Lesson 25 (Tues 4 Oct): Laplace transform (Section 7.1)
Lesson 26 (Wed 5 Oct): Quiz 06
Quiz 06
Your Quiz 6 has been graded.
Lesson 27 (Fri 7 Oct): Laplace transform and Inverse Laplace transform (Sections 7.1 and 7.2.1)
Review Laplace transforms.
Use the table (handout) to compute inverse Laplace transforms.
What is the inverse Laplace transform of 1? It's the Dirac delta function (unit impulse function).
Lesson 28 (Tues 11 Oct): Solving differential equations using linear transforms (Section 7.2.2)
Compute the Laplace transform of f'(t), f''(t), f'''(t), etc.
Solve a differential equation with initial conditions using Laplace tranforms.
Lesson 29 (Wed 12 Oct): First translation theorem (Section 7.3.1)
Quiz 07
Compute the Laplace transform of e^{at} f(t).
Announcement (Thurs 13 Oct):
Quiz 7 has been graded.
Lesson 30 (Fri 14 Oct): First translation theorem (Section 7.3.1)
Lesson 31 (Mon 17 Oct): Unit step function (Section 7.3.2)
Unit step function U(t-a)
Laplace transform of f(t-a)U(t-a) equals e^{-as}F(s), where F(s) = Laplace transform of f(t).
Lesson 32 (Tues 18 Oct): Unit step function in a differential equation (Section 7.3.2) and Review
Lesson 33 (Wed 19 Oct): Review
Lesson 34 (Fri 21 Oct): Test 2
There are 8 problems on your Test 2, spread out over 6 pages. Some problems have multiple parts.
The cutoff date for your 12-week grades is today. Your 12-week grades will be submitted immediately after I have completed grading your exams.
Lesson 35 (Mon 24 Oct): Derivative of linear transform, convolution (Section 7.4)
Compute d/ds(F(s)), where F(s) is the Laplace transform of f(t).
Derive Laplace transform of t^n f(t).
Discuss the definition of convolution f*g.
Derive the Laplace of the convolution f*g: (Laplace of f*g) = (Laplace of f)(Laplace of g).
Lesson 36 (Tues 25 Oct): Dirac delta function (Section 7.5)
Compute the inverse Laplace for F(s)G(s) using convolution.
Your Test 2 has been graded. It will be passed back tomorrow.
Test 2 average: 82.5121, standard deviation: 11.6225. The highest grade on this test is 100 out of 100 points.
Lesson 37 (Wed 26 Oct): Dirac delta function and Matrices, basic definitions (Section 7.5 and Appendix B1)
No quiz today.
Discuss why the Laplace transform of the Dirac delta function δ(t-t_0) equals e^{-t_0 s} (we proved this in class today!).
Go through basic properties of matrices.
Lesson 38 (Fri 28 Oct): Gauss or Gauss-Jordan elimination (Appendix B2)
If det(A) ≠ 0, where A is a square matrix, then A is nonsingular.
If det(A) = 0, where A is a square matrix, then A is singular.
Theorem: An n x n matrix A has a multiplicative inverse A^{-1} ⇔ A is nonsingular.
Formula for the inverse of a matrix via using cofactors.
Learn how to construct an augmented matrix. Learn how to do elementary row operations and put the matrix into row-echelon (REF, Gauss) and reduced row-echelon form (RREF, Gauss-Jordan).
3rd period substitute: Professor Ana Maria Soane
4th period substitute: Professor Van Nguyen
5th period substitute: Professor Kostya Medynets
Lesson 39 (Mon 31 Oct): Solving linear systems by Cramer's rule (Handout)
Case 1: Given x - y + z = 1, 2x + z = 0, 3y - z = 2, use RREF to solve for x, y, z (you should get the unique solution; the solution is a point in R^3).
Case 2: Given x - y + z = 1, 2x + z = 0, -2x - z = 0, use RREF to solve for x, y, z (you should get infinitely-many solutions; the solution forms a line in R^3).
Case 3: Given x - y + z = 1, 2x + z = 0, 2x + z = 1, use RREF to solve for x, y, z (the system of equations is inconsistent, so there is no solution; the solution is the empty set in R^3).
Use Cramer's Rule to solve for x, y, z in Case 1 (notice that this technique works only if the system of equations has a unique solution, i.e., when writing the system of equations as a product of matrices Ax = b, then the matrix A must be invertible).
Lesson 40 (Tues 1 Nov): Eigenvalues and eigenvectors (Appendix B.3)
If A is invertible (nonsingular, i.e., det(A) ≠ 0), find A^{-1} using an augmented matrix and RREF.
Find eigenvalues and eigenvectors using RREF.
Lesson 41 (Wed 2 Nov): Solving systems of differential equations using linear transforms (Section 7.6)
Quiz 08
Use Laplace transform to solve x'(t) = - x + y and y'(t) = 2x given the initial conditions x(0) = 0 and y(0) = 1.
See Notes for Section 7.6 (pages 1 - 3).
How do you check that your solution to a system of differential equations is correct? See the Mathematica file for some examples.
Announcement (Thurs 3 Nov):
Quiz 8 is currently being graded.
Lesson 42 (Fri 4 Nov): Electrical networks and Systems with real, distinct eigenvalues (Sections 3.3, 7.6, 8.2.1)
Kirchhoff's First Law: at each branch point, the sum of the currents flowing in equals the sum of the currents flowing out. For example, i_1 = i_2 + i_3.
Using Kirchhoff's Second Law (the total voltage drop is equal to the voltage drop across each part of the loop), write differential equations associated to more generalized electrical circuits.
Solve for the currents, for example, i_1 and i_2.
When given two distinct real eigenvalues, write down two solution vectors to X' = AX, where X is the 2x1 column matrix [x(t) y(t)]^t.
When given two distinct real eigenvalues, write down the general solution vector to X' = AX. This general solution vector is also called complementary solution vector.
See Notes for Section 8.2.1 (pages 4 - 7). Why does the complementary solution vector look like this? We'll talk about this when I'm back on Monday 14 Nov.
What about a particular solution vector? How do you find it? Wait until Monday 14 Nov!
A copy of the message on Blackboard about today's class:
The most important aspect of today's class is if I give you any electrical network, you have to know how to set up a system of differential equations (don't forget the initial conditions). Then we solved for currents i1 (which is a function of t) and i2 (which is a function of t) by converting the two differential equations into two algebraic equations involving I1 = L(i1) and I2 = L(i2) by applying the Laplace transform to every term. Once you separated I1 and I2 into separate equations and they look "nice", you used the inverse Laplace transform to find the currents i1 and i2.
Is there another way to solve a system of differential equations besides using the Laplace transform?
Yes, but only in special cases. That is, given general homogeneous linear first-order system, i.e., X' = AX (and no initial conditions), the solution vectors involve eigenvalues and eigenvectors. We'll go through this again on Monday 14 Nov, and we'll talk about why solution vectors X1 and X2 look like this on an interval I. Then what does the general solution vector look like? It's a linear combination of all the solution vectors via the superposition principle (we call such general solution vector complementary solution vector).
Note: can we find the solution vectors to X' = AX + F? Yes. We have to find the complementary solution vector Xc and then a particular solution vector Xp. Then the general solution vector is of the form X = Xc + Xp. We'll talk about this on Monday 14 Nov.
How do you check that your solution to a system of differential equations is correct? See the Mathematica file for some examples.
Announcement (Sun 6 Nov):
Quiz 8 has been graded.
Lesson 43 (Mon 7 Nov): Linear systems of differential equations (Section 8.1)
Go through one example with the substitute and then a midshipman does a similar example at the board.
Spend the rest of the class working through WebAssign assignments individually or in small groups.
If anyone is off task during class (staring at stock prices, playing games from your phone or computer, reading the news, sending or reading text messages, working on homework for another class, studying for another class, etc.), then the professor will write down your name and you will lose points on Class Participation. Resting and downtime are important, but do this at your own time, not when you are surrounded by 20+ midshipmen who are sitting next to you to learn differential equations.
3rd period substitute (0955-1045): Professor Carolyn Chun
4th period substitute (1055-1145): LT Ryan P. Bailey
5th period substitute (1330-1420): Professor Max Wakefield
Lesson 44 (Tues 8 Nov): Linear systems of differential equations (Section 8.1)
Spend the class time working through WebAssign assignments individually or in small groups.
Remember to go at your own pace and actively work through these problems since you have Test 3 coming up.
There are more practice problems in the textbook (the odd numbered problems have solutions in the back of the book so that you may check your solution).
If you are still stuck and need help, we have plenty of resources for help (30 hours of free Math Lab, 40+ hours of private tutoring at the Academic Center, 2 hours of MGSP, peer-tutoring with MIDN Karl Florida and MIDN Maximus Lear, working in small groups with other midshipmen), and the course textbook (solutions to odd numbered problems are in the back of the book).
How do you check that your solution to a system of differential equations is correct? See the Mathematica file for some examples.
If anyone is off task during class (staring at stock prices, playing games from your phone or computer, reading the news, sending or reading text messages, working on homework for another class, studying for another class, etc.), then the professor will write down your name and you will lose points on Class Participation. Resting and downtime are important, but do this at your own time, not when you are surrounded by 20+ midshipmen who are sitting next to you to learn differential equations.
3rd period substitute (0955-1045): Professor Irina Popovici
4th period substitute (1055-1145): LT Ryan P. Bailey
5th period substitute (1330-1420): Professor Max Wakefield
Lesson 45 (Wed 9 Nov): Systems with real, distinct eigenvalues (Section 8.2.1)
Quiz 09 (to be given at the beginning of the class; no external resources are allowed except the Laplace transform table).
Quiz 09: remember that if there are odd number of midshipmen in the class, then one person has to work alone. A group of 3 or more is NOT allowed.
See Notes for Section 8.2.1 (pages 4 - 7).
Spend the rest of the class working through WebAssign assignments individually or in small groups.
How do you check that your solution to a system of differential equations is correct? See the Mathematica file for some examples.
If anyone is off task during class (staring at stock prices, playing games from your phone or computer, reading the news, sending or reading text messages, working on homework for another class, studying for another class, etc.), then the professor will write down your name and you will lose points on Class Participation. Resting and downtime are important, but do this at your own time, not when you are surrounded by 20+ midshipmen who are sitting next to you to learn differential equations.
3rd period substitute (0955-1045): Professor Darren Creutz
4th period substitute (1055-1145): LT Ryan P. Bailey
5th period substitute (1330-1420): Professor Max Wakefield
Lesson 46 (Mon 14 Nov): Systems with complex eigenvalues (Section 8.2.3)
We'll review systems with real, distinct eigenvalues (Section 8.2.1) first. Why does the complementary solution vector look like this, via a linear combination of eigenvalues and eigenvectors? We'll discuss this today.
What about a particular solution vector? How do you find it? We'll talk about this today!
If the system of differential equations X' = AX +F has two real, distinct eigenvalues, where A is a 2x2 matrix, then how does the general solution look like? If X_c is the complementary solution vector and X_p is a particular solution vector, then X is the 2x1 column vector X = X(t) = X_c + X_p.
Now, what if the eigenvalues include complex conjugate numbers? Today!
Due to a tanker carrying propane overturning on the Severn River bridge this morning at 5:15 am, a 6 minute drive to the Academy took 3 hours. So you were with a fantastic professor with morning (and a shout-out to LCDR Jeff Lineberry for willing to assist both classes)!
3rd period substitute (0955-1045): Professor Irina Popovici
4th period substitute (1055-1145): Professor Irina Popovici
I have completed grading your Quiz 9. They will be returned tomorrow.
Lesson 47 (Tues 15 Nov): Systems with real, distinct eigenvalues and Systems with complex eigenvalues (Sections 8.2.1 and 8.2.3)
Lesson 48 (Wed 16 Nov): Review
Lesson 49 (Fri 18 Nov): Test 3
Each problem is worth 10 points.
Jump around and solve the problems that you do know how to complete.
You don't have to write out ALL the details to each problem; if you know any tricks, you are free to use them.
For example, you don't always need to expand a term like Y(s^2 + 1)(s^2 + 1). Write this as Y(s^2 + 1)^2.
As another example, if you want to solve for X and Y from the system of equations 3X + 4Y=210s and 3X - 4Y = 120/s, add or subtract the two equations; don't solve for one variable and then substitute into the other.
Pace yourselves.
Announcement (Sun 20 Nov):
Test 3 has been graded. It will be passed back tomorrow.
Your Test 3 average is 88.1207 out of 100 points with standard deviation 9.9755. The highest grade is 99.5.
Lesson 50 (Mon 21 Nov): Fourier series (Section 11.2)
Wed 7 Dec (at 2359) is the last day that I am accepting LATE WebAssign assignments. You can continue to work on WebAssign assignments after that day for zero credit.
Lesson 51 (Tues 22 Nov): Even and odd functions (Section 11.3)
Announcement (Tues 22 Nov):
For those making up Test 3 during lunch, please go to Room CH320.
If you want to start on your make-up test early, then please arrive early. I will be in my office starting at 1200.
Lesson 52 (Wed 23 Nov): Half-range expansions (Section 11.3)
Early schedule today.
3rd period: 0855-0945
4th period: 0955-1045
5th period: 1055-1145
Quiz 10
Happy Thanksgiving!
Lesson 53 (Mon 28 Nov): Separable partial differential equations (first and second order) (Section 12.1)
Lesson 54 (Tues 29 Nov): Midshipmen-driven review
Spend the class time working through WebAssign assignments individually or in small groups.
Practice Section 12.1 problems individually or in small groups.
3rd period substitute (0955-1045): CDR Kirsten Davis
4th period substitute (1055-1145): CDR Kirsten Davis
5th period substitute (1330-1420): Professor Caroline Melles
Lesson 55 (Wed 30 Nov): Midshipmen-driven review
Quiz 11
Spend the rest of the class working through WebAssign assignments individually or in small groups.
3rd period substitute (0955-1045): Professor Caroline Melles
4th period substitute (1055-1145): CDR Thomas del Zoppo
5th period substitute (1330-1420): CDR Thomas del Zoppo
Lesson 56 (Fri 2 Dec): Heat equation (zero ends) (Section 12.3)
Lesson 57 (Mon 5 Dec): Review
Let's redo Quiz 11 together as a class. Then review all day today.
Lesson 58 (Tues 6 Dec): Test 4
Systems with complex eigenvalues (Section 8.2.3) will be on the test.
There are 5 problems, and some problems have multiple parts.
Pace yourself. Skip around and work on those you know how to solve first.
The comment from this morning that another professor (my colleague) is dropping the lowest Test grade is just a rumor. I confirmed this with that professor, and we are NOT dropping the lowest test grade.
Heat equations will be on Test 4, as well as on the Final Exam.
Lesson 59 (Wed 7 Dec): Review for final exam
Today at 2359 is the last day that I am accepting LATE WebAssign assignments. You can continue to work on WebAssign assignments after today for zero credit.
SOF during class today!
A small party to celebrate your successes from this semester!
Go through some problems from Test 4 and go through practice Final Exams together as a class.
Announcement (Thurs 8 Dec):
I will grade Test 4 most of today.
Lesson 60 (Fri 9 Dec): Review for final exam
Due to the Army-Navy game, I will give you time on Wednesday to complete your SOF.
Also because of the football game and since many midshipmen may be absent today, the party has been moved to Wednesday.
Test 4 will be returned to you today during class.
Update: several midshipmen have received 100% on Test 4. There are MANY midshipmen who received an A on Test 4.
Test 4 average: 76.56%. This is a solid C.
Go through practice Final Exams together as a class.
Do as many practice problems today, including deriving the heat equation with zero ends.
Announcement (Sun 11 Dec):
MIDN Karl Florida is hosting a 2-hour MGSP review session in CH155 tonight: 2000 - 2200.
Announcement (Mon 12 Dec):
The Academic Center is conducting a final exam review session in CH175: 1500 - 1600.
Professor Im has a certain appointment until 1600. Professor Im's walk-in EI in her office: 1600 - 1815.
If her appointment ends sooner, she will aim to be in her office starting at 1600.
Final Exam (Tues 13 Dec): 1300 - 1600
Section 3021: Room CH131
Section 4021: Room CH155
Section 5021: Room CH132
The exam will consist of a 20-question Multiple Choice section and a Free Response section distributed on paper. The multiple choice part will use a scantron and the answers need to be bubbled in. For the free response, space will be provided on the paper for the answers.
The TI-36X Pro calculator is allowed. Calculators may NOT be shared. The exam is comprehensive. The sections are equally weighted. In addition to your calculator, please bring #2 pencils, pens, an eraser and nothing else. Leave everything else outside the examination room.
A Laplace/Fourier/integration Table (the table you have used all semester) will be provided and no other notes will be allowed. So leave your Laplace table and everything else in the hallway; do not bring any paper to the exam room-- plenty of paper will be provided.
Given specific parameters k, L = length of metallic rod, and f(x) = u(x,0), you have to know how to derive u(x,t) from the linear second-order partial differential equation k u_{xx} = u_t by considering all three cases: λ = 0, λ < 0, and λ > 0. In other words, you must show the details to all three cases, not just plug and chug into the heat equation u(x,t).
I'm bringing homemade triple chocolate fudge brownies, with chocolate chips, chocolate chunks, dark Dutch cocoa, and cream cheese frosting.
Announcement (Mon 19 Dec):
Your Final Exam average for the Multiple Choice portion: 74.127%.
Your Final Exam average for the Free Response portion: 80.381%.
Have a wonderful break!
Series electrical circuits