Math 415: Differential Equations
Basic Information
This class provides a basic introduction to ordinary differential equations, Fourier series,
and separation of variables in partial differential equations. The text we will be using is
Elementary Differential Equations and Boundary Value Problems, by Boyce and
Prima (7th edition, abriged).
Grading:
There will be two midterms and a final. In addition, there will be weekly quizzes (Thursdays,
during recitation). The final grade will be based on a weighted average of your grades on:
quizzes (20%), Midterm I (20%), Midterm II (20%), and the final exam (40%).
The first midterm will be in class, on Wednesday Oct. 11th, and the second midterm will
be in class, on Wednesday, Nov. 8th.
The final exam for the 9:30am class will be on Wednesday, Dec.
6th, from 7:30-9:18am. The exam will be given in our regular lecture room: RA 115.
The final exam for the 2:30pm class will be
on Thursday, Dec. 7th, from 11:30-1:18pm. The exam will be given in our
regular lecture room: KN 190.
Reaching me:
My office is in MW 510 (Math Tower). Office hours are MWF from 10:30-11:30, or by appointment.
If you need to contact me, the most efficient way is via e-mail at jlafont@math.ohio-state.edu
Additional Material
In the last week of classes, we started discussing systems of linear
differential equations. You can now download the following
additional material (solutions and notes will be posted later
today):
Recommended Homework Problems:
The following problems will not be collected. However, the weekly
quiz will be closely modelled on the recommended homework problems.
- Section 2.1: 15, 16, 17, 18, 19.
- Section 2.2: 1, 2, 3, 4, 9, 10, 11, 12, 31, 32, 33.
- Section 2.3: 2, 4, 10, 18, 21, 24, 30.
- Section 2.4: 3, 4, 5, 6, 13, 14, 23, 24, 28, 29, 30, 31.
- Section 2.5: 2, 3, 7, 8, 9, 22, 26.
- Section 2.6: 1, 2, 3, 4, 5, 6, 7.
- Section 3.1: 5, 6, 7, 8, 13, 14, 15, 16, 20, 21, 28, 29, 30, 34, 35, 36.
- Section 3.2: 4, 5, 6, 7, 8, 9, 13, 14, 21, 23.
- Section 3.3: 5, 6, 7, 8, 15, 16, 19.
- Section 3.4: 13, 14, 15, 16, 17, 18, 27, 28.
- Section 3.5: 7, 8, 9, 10, 11, 12, 20, 23, 24, 25, 26.
- Section 3.6: 1, 2, 3, 4, 13, 14, 16, 17.
- Section 3.7: 1, 2, 5, 7, 13, 14, 15, 16, 29, 30.
- Section 5.1: 1, 2, 9, 10, 13, 14, 17, 21, 22.
- Section 5.2: 1, 2, 3, 4, 5, 6, 20.
- Section 5.3: 2, 3, 6, 7, 10, 11, 12.
- Section 3.8: 3, 4, 6, 7, 11, 13, 17.
- Section 3.9: 5, 6, 9, 10, 11, 15.
Topics for the first exam:
Here is a list of things you should be able to do:
- Recognize and solve separable differential equations dy/dx = F(x)/G(y). This involves
separating your variables and integrating. You should also be able to recognize homogenous
differential equations (of the form dy/dx = F(y/x)), and be able to apply the substitution v= y/x.
- Recognize and solve linear equations dy/dx + P(x) y = Q(x). This involves multiplying
by the integrating factor, recognizing that the left hand side is the derivative of a product, and
integrating.
- Recognize and solve exact differential equations M(x,y) + N(x,y) dy/dx = 0. This involves
finding a function (of x,y) with prescribed partial derivatives.
- Know the two basic theorems concerning existence and uniqueness of differential equations.
The theorems deal with (1) differential equations of the form dy/dx + P(x) y = Q(x), and (2) with
differential equations of the form dy/dx = F(x,y).
- Know how to do a qualitative analysis for autonomous differential equations. These are
equations of the form dy/dx = F(y). In particular, finding and classifying equilibrium solutions, and
being able to identify the general behavior of solutions to the differential equation.
- Know how to solve word problems. The three step process is (1) translating the word
problem into a differential equation, (2) solving the differential equation, and (3) translating back
the mathematical answer into information on the physical problem you are dealing with.
Topics for the second exam:
Here is a list of things you should be able to do:
- Know the general results concerning the existence/uniqueness of solutions to differential
equations of the form y'' + p(x) y' + q(x) = g(x).
- Know the definition of the Wronskian, how it relates to finding all solutions of a differential
equation of the above form. Know the statement of Abel's theorem, and how to use it to find a
second solution to a differential equation, given a first solution.
- Know how to use the reduction of order method to find a second solution to a differential equation,
given a first solution. This boils down to trying to find a second solution of the form v(x)y(x), where y(x)
is your first solution, and v(x) is a function you need to solve for.
- Know how to solve differential equations of the form y'' + py' + qy = 0, where p, q are constants.
This boils down to
finding the roots of the characteristic polynomial, and knowing how to obtain solutions depending
on the nature of the roots (distinct real roots, complex roots, or repeated real roots).
- Know how to use the method of undetermined coefficients to find a solution to a differential
equation of the form y'' + py' + qy = g(x), where p, q are constants and g(x) is a function. This involves
making a guess as to the form of the solution, usually by trying a combination of the function g(x) and it's
various derivatives. By matching coefficients, you try to solve for the precise combination that will
give you a solution.
- Know how to use the variation of parameters method to find the solution to a differential
equation of the form y'' + py' + qy = g(x), where p, q are constants and g(x) is a function. This involves
trying to find a solution of the form v(x) Y(x) + w(x) Z(x), where Y, Z are solutions to the homogenous
differential equation y'' + py' + qy = 0, and v, w are functions you need to solve for.