Math 0290 Schedule and Practice Problems (updated 8/19, pdf, doc)
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This schedule may fluctuate slightly especially as we figure out how to accommodate your safety and health. Currently there are 13 problem sets (including HW 0) due throughout the semester. This may decrease in number as I determine the best way to deal with the homework assignment near midterm 2, but definitely not increase.
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Homework for the week is listed in red under description of the material covered that week along with the date it is due. These assignments are to be submitted on Canvas Assignments. Lectures are listed by day including problems you can practice from that section to better understand the material (some of these problems are the homework problems for the following week). Then there are learning objectives for the week. They are split up into many smaller items, so it seems like a lot, but many are not time consuming. You can use this list to make sure you understood everything from the week and review for midterms and exams. *Learning objectives that are surrounded by stars won’t be tested on the exam but still maybe useful things to know outside of this course. *
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Week 1: Introductions to the course
Homework #0 Due Monday August 24 by midnight
Answer questions about Syllabus

What do you like about the syllabus?

What would you change about the syllabus?

What are you excited about in this course?

What are you nervous about in this course?

Any additional comments?

August 19: Welcome, Introduction to Differential Equations

Practice problems Chapter 1.1


August 21: First Order Initial Value Problems

Practice problems Chapter 2.1 #36, 1015, 2128.

Students should be able to:

Explain what a differential equation is

Identify what is required for a differential equation

Give examples of some systems that use differential equations


Identify a first order differential equation

Write an ordinary differential equation (ODE) in normal form

Check a given solution for an ODE

Find a particular solution for an ODE in an initial value problem

Week 2: Intro to Numerics
Homework #1 due Friday August 28 by midnight:
Chapter 1.1 #1, 5, 7, 11
Chapter 2.1 #1, 3, 5, 12, 13, 15

August 24: Numerical Methods. Euler’s Method, Computer tools including Matlab for Differential Equations

6.1 Practice problems #19, 11


August 26: Numerical Methods. RungeKutta Methods

6.2 Practice problems #19


August 28: Numerical Methods. Numerical Error

6.3 Practice problems #16, 1113

Students should be able to:

Explain what a numerical solution is

Apply the Euler’s Method for modeling differential equations with initial value problems (IVP)

Compute a few steps of the Euler’s Method for an IVP

Calculate the error from using Euler’s Method compared to an exact solution

Explain the difference between truncation error and roundoff error

*Apply Euler’s Method for a system of differential equations*


Apply the second order RungeKutta Method for modeling differential equations

Calculate the error for the second order RungeKutta Method


Apply the fourth order RungeKutta Method for modeling differential equations

Calculate the error for the fourth order RungeKutta Method


*Use Matlab to apply Euler’s Method*

*Use Matlab to apply the 2nd order RungeKutta Method*

*Use Matlab to apply the 4th order RungeKutta Method*

*Plot Matlab results*
Week 3: Separable Equations and modeling
Homework #2 due Friday September 4th by midnight:
Chapter 6.1 #3, 5
Chapter 6.2 #23
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August 31: Separable Equations

2.2 Practice problems #122, 2329, 3335


September 2: Models of Motion

2.3 Practice problems #110


September 4: First Order Linear Equations

2.4 Practice problems #121, 29

Students should be able to:

Explain what a separable equation is

Rewrite and solve a separable equation

Describe the case when you cannot solve a separable equation

Explain what a general solution is

Explain what an implicitly defined solution is and the difference between that and an explicitly defined solution


Solve for an equation of motion using physical models

Apply the scaling variables method to simplify separation of variables

Explain what a homogeneous linear equation and an inhomogeneous equation are

Explain what the coefficients of the equation are

Find the solution to a homogeneous equation

Find the solution to an inhomogeneous equation

Week 4: Linear firstorder equations and Intro to Second order equations
Homework #3 due Friday September 11th by midnight:
Chapter 2.2 #3, 5, 9, 33
Chapter 2.3 #9
Chapter 2.4 #5, 15, 19
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September 7: Mixing Problems

2.5 Practice problems #17, 9, 10


September 9: Electrical Circuits

3.4 Practice problems #119


September 11: Second Order Equations

4.1 Practice problems #120, 2630

Students should be able to:

Set up differential equation for a mixing problem

Identity the volume rates and concentration in a mixing problem

Solve a mixing problem differential equation


Explain the component laws (Ohm’s law, Faraday’s law, capacitance law)

Explain Kirchhoff’s voltage law

Explain Kirchhoff’s current law

Derive a differential equation for a circuit using the component laws and Kirchhoff’s laws

Solve a differential equation derived for a circuit using given variables and initial conditions


Identify a second order differential equation

Explain how a forcing term affects a linear differential equation

Derive a second order differential equation for a springmass equilibrium (SME) system with an external force

Solve for the spring constant in the SME system

Solve the second order differential equation for a SME system when the system is undamped

Solve for the period of the SME system

Solve for the angular frequency of the SME system

Solve for the numerical frequency of the SME system

Define what linear combination is

Define what linearly dependent and linearly independent are

Find a fundamental set of solutions to a second order differential equation

Determine the Wronskian of two solutions of a second order differential equation

Explain the independence of two solutions using the result of the Wronskian calculation


Week 5: Second order equations and harmonic motion
Homework #4 due Friday September 18th by midnight:
Chapter 2.5 #5, 9b
Chapter 3.4 #1, 3, 5, 7, 11
Chapter 4.1 #1, 3, 9, 17
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September 14: Linear Homogeneous Equations with Constant Coefficients

4.3 Practice problems #136


September 16: Linear Homogeneous Equations with Constant Coefficients (con’t)

4.3 Practice problems #136


September 18: Harmonic Motion

4.4 Practice problems #112, 1416, 18

Students should be able to:

Derive the characteristic equation of a differential equation

Solve for the characteristic root of a differential equation

Determine if the roots are two distinct real roots, two distinct complex roots, or one repeated root

Use the characteristic roots to determine the general solution of the differential equation with two distinct real roots

Find a particular solution using a set of initial conditions with two distinct real roots


Use the characteristic roots to determine the general solution of the differential equation with two distinct complex roots

Find a particular solution using a set of initial conditions with two distinct complex roots


Use the characteristic roots to determine the general solution of the differential equation with one repeated root

Find a particular solution using a set of initial conditions with one repeated root



Identify the equation for simple harmonic motion (SHM)

Derive the characteristic equation for SHM

Find the characteristic roots for SHM

Solve for the solution for SHM

Plot a particular solution for the SHM case

Rewrite the solution to the SHM case using polar form

Identify the amplitude of oscillations for a SHM

Identify the phase of oscillations for a SHM


Identify the equation for damped harmonic motion (DHM)

Derive the characteristic equation for DHM

Find the characteristic roots for DHM

Solve for the solution for DHM

Plot a particular solution for the DHM case

Rewrite the solution to the DHM case using polar form

Identify the amplitude of oscillations for a DHM

Identify the phase of oscillations for a DHM

Week 6: Inhomogeneous second order equations
Homework #5 due Friday September 25th by midnight:
Chapter 4.3 #1, 9, 17, 35
Chapter 4.4 # 1, 7

September 21: Inhomogeneous Equations. Undetermined Coefficients

4.5 Practice problems #129


September 23: Undetermined Coefficients (continued)

4.5 Practice problems #129


September 25: Inhomogeneous Equations. Variation of Parameters

4.6 Practice problems #110

Students should be able to:

Identify the form of an inhomogeneous linear equation

Find the general solution to the inhomogeneous equation

Find the particular solution to the inhomogeneous equation


Explain the method of undetermined coefficients

Identify the case when the method of undetermined coefficients can be applied to differential equation

Apply the method of undetermined coefficients to the case with exponential forcing terms

Apply the method of undetermined coefficients to the case with trigonometric forcing terms

Apply the complex method to the case with trigonometric forcing terms


Apply the method of undetermined coefficients to the case of polynomial forcing terms

Identify exceptional cases when the method of undetermined coefficient cannot be used

Apply the method of undetermined coefficients to the case of a linear combination of forcing terms (exponential, trigonometric, polynomial)


Apply the technique of variation of parameters to find a particular solution to a second order differential equation
Week 7: Forced harmonic Motion, Laplace Transforms
Midterm #1 will be available from Tuesday, September 29th 12:00pm to Thursday October 1st 11:59am, covering material up to and including HW4

September 28: Forced Harmonic Motion

4.7 Practice problems #311


September 30: Midterm 1, no class

October 2: Laplace Transform

5.1 Practice problems #129

Students should be able to:

Apply the technique of undetermined coefficients to analyze harmonic motion with an external sinusoidal forcing term with no damping

Explain what the driving frequency of an equation of harmonic motion with external sinusoidal forcing is

Determine how the driving frequency compares to the natural frequency

Find the particular solution when the driving frequency does not equal the natural frequency

Find the general solution to the inhomogeneous equation

Plot the solution to harmonic motion with an external sinusoidal forcing term

Explain what beats are

Calculate the mean frequency and half difference


Find the particular solution when the driving frequency equals the natural frequency

Find the general solution to the inhomogeneous equation

Explain what resonance is



Apply the technique of undetermined coefficients to analyze harmonic motion with an external sinusoidal forcing term with damping

Determine the transfer function

Determine the gain

Determine the transient term and the steady state term


Define a Laplace Transform and state why it is useful

Apply a Laplace transform to an exponential function

Apply a Laplace transform to a linear function

Apply a Laplace transform to a sinusoidal function

Apply a Laplace transform to piecewise continuous functions

Week 8: Beginning Laplace Transforms
Homework #6 due Friday October 9th by midnight:
Chapter 4.5 #1, 5, 11, 15, 19
Chapter 4.6 #1, 3, 5
Chapter 4.7 #3, 11

October 5: Laplace Transform. Basic Properties

5.2 Practice problems #141


October 7: The Inverse Laplace Transform

5.3 Practice problems #136


October 9: Using the Laplace Transform to Solve Des

5.4 Practice problems #126

Students should be able to:

Define a Laplace transform on derivatives

Apply a Laplace transform on piecewise differentiable and continuous functions


Recognize and apply properties of the Laplace transform

Apply the linear property of a Laplace transform

Apply the exponential property of a Laplace transform

Apply the derivative property of a Laplace transform


Find the inverse Laplace transform

Compute inverse Laplace transforms using a table

Compute the partial fraction decomposition for a rational function

By applying the coefficient method

By apply the substitution method


Compute the inverse Laplace transform of rational functions


Use the Laplace transform to solve differential equations

Use the Laplace transform to solve the initial value problem for a homogeneous differential equation

Use the Laplace transform to solve the initial value problem for a inhomogeneous differential equation

Use the Laplace transform to solve a higher order equation

Week 9: Various functions to help with Laplace Transforms
Homework #7 due Friday October 16th by midnight:
Chapter 5.1 #7, 13, 15, 29
Chapter 5.2 #5, 11, 19, 29
Chapter 5.3 #3, 7, 11, 19

October 12: Discontinuous Forcing Term

5.5 Practice problems #125


October 14: Student SelfCare Day (no classes)

October 16: The Dirac Delta Function

5.6 Practice problems #19

Students should be able to:

Define the Heaviside function

Rewrite piecewise differentiable function in terms of the Heaviside function


Use the Heaviside function to find the Laplace transform of a piecewise differential function

Use the Heaviside function to find inverse Laplace transform of a function

Solve an initial value problem with piecewise defined forcing function

Compute the Laplace transform of the periodic function the square wave

Define what impulse is in terms of physical concepts

Define the delta function

Calculate the Laplace transform of a delta function


Find the unit impulse response function for a differential equation
Week 10: Laplace Transform (cont), Systems of differential equations
Homework #8 due Friday October 23rd by midnight:
Chapter 5.4 #7, 11, 21
Chapter 5.5 #1, 3, 11, 17
Chapter 5.6 #2, 3, 5, 7

October 19: Convolutions

5.7 Practice problems #424


October 21: Introduction to Systems

8.1 Practice problems #116


October 23: Systems (cont)

8.2 Practice problems #16, 1316

Students should be able to:

Define a convolution of two functions

Calculate the Laplace transform of a convolution

Find a solution to the general initial value problem using the Laplace transform of a convolution

Apply the property of a convolution of a piecewise continuous function and a delta function

Apply the property of a derivative of a convolution of two piecewise continuous functions


Identify what makes a model nonlinear

Identify what makes a model autonomous

Identify the susceptibleinfectedrecovered (SIR) model

Write a system of equations in vector notation

Determine the dimension of a system of firstorder equations

Define a planar system

Reduce a higherorder equation to a system of first order equations


Identify the predatorprey/LotkaVolterra system

Plot the solutions to the LoktaVolterra system dependent on time

Plot the parametric curves of the solution of the LotkaVolterra model

Define phase plane, and phase plane plot/solution curve

Identify the phase space of a system of ndimensions

Plot the direction field of a planar system

Week 11: Systems of differential equations, Constant coefficients
Homework #9 due Friday October 30th by midnight:
Chapter 5.7 #6, 8, 10
Chapter 8.1 #5, 7, 13, 15
Chapter 8.2 #11, 13, 15 (use pplane.jar)
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October 26: Systems (cont)

8.3 Practice problems #16


October 28: Linear Systems with Constant Coefficients

9.1 Practice problems #18, 1623


October 30: Planar Systems

9.2 Practice problems #127, 5861

Students should be able to:

Show an IVP has a unique, welldefined solution

Explain why two solution curves in phase space for an autonomous system cannot meet at a point unless the curves coincide


Explain what a nullcline of a system of autonomous equations is

Find the nullclines of a system of autonomous equations

Plot the nullclines of a system of autonomous equations

Find an equilibrium point of a system of autonomous equations

Determine the equilibrium solution

Explain what an equilibrium solution is


Explain what an eigenvalue of a matrix means

Explain what an eigenvector of a matrix is

Find the eigenvalues and eigenvectors of a matrix

Use eigenvalues and eigenvectors to find a fundamental set of solutions to a system of differential equations

Determine that solutions of planar systems are linearly independent

Use real eigenvalues and eigenvectors to find a general solution of a planar system

Use complex eigenvalues and eigenvectors to find a general solution of a planar system

Use one real eigenvalue with multiplicity of 2 to find a general solution of a planar system
Week 12: Nonlinear Systems
Midterm #2 will be available from Tuesday, November 3rd 12:00pm to Thursday November 5th 11:59am covering material up to and including HW8
Homework #10 due Friday November 6th by midnight (to be changed)
Chapter 8.3 #1, 3, 5
Chapter 9.1 #3, 5, 17, 19
Chapter 9.2 #13, 13, 15, 59

November 2: Phase Plane Portraits

9.3 Practice problems #123


November 4: Midterm #2, no class

November 6: Nonlinear Systems: Equilibria, Linearization

10.1 Practice problems #116

Students should be able to:

Find the three types of solutions when there are two real distinct eigenvalues

Find the exponential solutions/halflife solutions

Determine what eigenvalues make the solution exponential

Determine what makes a solution unstable

Determine what makes a solution stable


Find saddle point solutions

Determine what eigenvalues makes the solution a saddle point

Explain what a saddle point of a planar system is

Explain what a separatrix is

Find the separatrix of a planar system


Find the nodal sink/nodal source solutions

Determine what eigenvalues make the solution nodal source/nodal sink

Explain what solutions look like



Find the three types of solutions when the eigenvalues are complex

Explain what solutions do when there is a center

Plot solution curves

Find the general solution


Explain what solutions do when there is a spiral sink

Plot solution curves

Find the general solution

Determine the direction of rotation of solutions


Explain what solutions do when there is a spiral source

Plot solution curves

Find the general solution

Determine if the direction of rotation of solutions



Explain what makes a differential equation nonlinear

Find equilibrium points of a nonlinear differential equation

Linearize the equation at the equilibrium point

Characterize equilibrium points (saddle point, nodal sink, nodal source, spiral sink, spiral source)

Week 13: Fourier Series
Homework #11 due Friday November 13th by midnight:
Chapter 9.3 #21
Chapter 10.1 #3, 7, 15

November 9: Fourier Series

12.1 Practice problems #122


November 11: Fourier Cosine and Sine Series

12.3 Practice problems #132


November 13:

12.4 Practice problems #111

Students should be able to:

Show that terms of the Fourier series are orthogonal

Find coefficients of terms in the Fourier series

Define piecewise continuous

Determine a Fourier series for a corresponding function

Determine if the Fourier series converge

Determine if a function is odd or even

Determine the Fourier series of an even function

Determine the Fourier series of an odd function

Use Euler’s formula to rewrite a real Fourier series in complex form

Determine the complex Fourier series from a corresponding function
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Week 14: Partial Differential Equations
Homework #12 due Friday November 20th by midnight:
Chapter 12.1 #5, 7, 13, 17
Chapter 12.3 #3, 7, 19, 31
Chapter 12.4 #3

November 16: Partial Differential equations

13.2 Practice problems #118


November 18: Review

November 20: Review
Students should be able to:

Solve the IVB for the heat equation

Explain what makes the boundary conditions homogeneous

Use separation of variables to rewrite a partial differential equation as two ODE

Set up the StrumLouiville problem

Explain what an eigenfuction is

Satisfy initial conditions of SL problem

Use a Fourier series to solve the SL problem

Find a steadystate temperature

Reading Day: Saturday November 21st
Final Exam:
Thanksgiving!
Students should be able to: REST! Enjoy the holidays! We made it!