MATH 2930:
Differential Equations for Engineers
Spring 2018
This is the web page for my three sections of MATH 2930 at Cornell in Spring 2018. I will post worksheets used in section (occasionally with answers included) here every week. I may also include outside material that I think is interesting and relevant to what we are doing in the course. You are not required to use anything on this web page, but I think you may find it useful. The main page for the course can be found here.
I want to thank Elliot Cartee and James Barnes for giving me access to their material from when they last taught this class.
My office hours are Wednesday 3:30-5:30pm in Thurston Hall 102.
Week 1: Wednesday 1/24 and Thursday 1/25
Learning goals
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Draw the direction field of a given differential equation.
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Use the direction field to make qualitative statements about solutions of differential equations, e.g. find equilibria and describe asymptotic behavior.
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Use these qualitative tools to determine whether a given differential equation is a suitable mathematical model for a specific problem.
Documents
start-of-semester information, Gradescope help, take-home syllabus quiz.
Hand in the take-home quiz on Gradescope by Sunday. Contact me if you have any trouble making an account or uploading the quiz.
Week 2: Wednesday 1/31 and Thursday 2/1
Learning goals
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Solve linear first order ODEs using integrating factors.
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Solve separable first order ODEs.
Documents
Below are some Desmos plots relevant for 2(3). The equation we get when we separate and integrate describes the red curve. It has different branches just like the graph y = tan(x). Naively solving the equation for y (using arctan) only gets us one branch (the green one). The constant solutions are plotted using purple dashed lines. You can play around with the value of the integration constant C using the slider on the left. You can also access these plots here.
Week 3: Wednesday 2/7 and Thursday 2/8
Learning goals
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Determine the stability of equilibria of autonomous ODEs.
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Use equilibria and stability to determine whether a given ODE is a good model for a given problem.
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Explore the consequences of the Existence and Uniqueness Theorem.
Not covered this week
Section 2.3: Modeling with First Order Differential Equations. There are plenty of problems in the book for you to do. Practice makes perfect!
Documents
Week 4: Wednesday 2/14 and Thursday 2/15
Learning goals
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Solve exact 1st order ODEs.
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Use an integrating factor to transform a 1st order ODE into an exact one and examine what the limitations of this method are.
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Solve homogeneous 2nd order ODEs where the characteristic polynomial has two distinct real roots.
Not covered this week
Section 2.7: Numerical Integration using Euler's Method.
Documents
Week 5: PRELIM 1
You can find some review problems below. I will post the solution to these problems on Tuesday.
Documents
Week 6: Wednesday 2/28 and Thursday 3/1
Week 7: Wednesday 3/7 and Thursday 3/8
Learning goals
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Find the general solution 2nd order ODE with constant coefficients when the driving term is a combination of polynomials, exponentials, sines and cosines.
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Deal with linear 2nd order ODEs where the driving term is a solution of the corresponding homogeneous equation.
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Given a solution of a homogeneous linear 2nd order ODE, use reduction of order to find another solution that is independent of the given one.
Documents
The graphs below are plots of solutions for Question 1. The red graph corresponds to the case where the frequencies are different. You can change the frequency of the driving term. Move around to see that the solutions explodes when both frequencies approach each other (here one of them is fixed at 3), but that the solutions are always bounded.
The green solution corresponds to the case where the frequencies are equal. Note that the solution is unbounded by moving around a bit.
Week 8: Wednesday 3/14 and Thursday 3/15
Learning goals
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Use variation of parameters to find a particular solution of a linear second order ODE when given two independent solutions of the homogeneous equation.
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Analyze properties of vibrating systems: free versus forced, damped versus undamped.
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Determine the steady state solution of an initial value problem.
Documents
Below I plotted the solution to the question where the driving term is piecewise-defined.
Week 9: Wednesday 3/21 and Thursday 3/22
Week 10: Wednesday 3/28 and Thursday 3/29
Learning goals
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Find the general solutions of a homogeneous linear higher order differential equation.
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Write down the system of equations describing a coupled system of masses and springs.
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Determine the eigenvalues and eigenfunctions of a boundary value problem.
Documents
Week 11: Wednesday 4/11 and Thursday 4/12
Learning goals
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Determine the eigenvalues and eigenfunctions of a boundary value problem.
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Compute the Fourier series of a periodic function.
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Analyze the convergence properties of a Fourier series.
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Match properties of a function to properties of its Fourier series.
Documents
Week 12: Prelim 2
You can find some review problems below. I will post the solutions here on Sunday.
Documents
Below you can play around with the number of terms in the Fourier series of the last problem. Note the slow rate of convergence and the Gibbs effect around the discontinuities.
Week 13: Wednesday 4/25 and Thursday 4/26
Learning goals
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Apply separation of variables to solve the heat equation with fixed (zero) temperature at both ends and given initial temperature distribution.
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Adapt the method so it applies to variations of the basic problem, e.g. insulated ends.
Documents
Week 14: Wednesday 5/2 and Thursday 5/3
Learning goals
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Solve wave propagation problems and Laplace equations with various boundary condi- tions.
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Derive a closed form for the solution of a general wave propagation problem in an infinite medium in terms of the initial conditions.
Documents