The results of these examples will be very useful for the rest of this chapter and most of the next chapter.

Welcome to my online math tutorials and notes. We will also define the even extension for a function and work several examples finding the Fourier Cosine Series for a function.

We also define the Laplacian in this section and give a version of the heat equation for two or three dimensional situations. We discuss classifying equilibrium solutions as asymptotically stable, unstable or semi-stable equilibrium solutions.

Bernoulli Differential Equations — In this section we solve linear first order differential equations, i. A unit circle completely filled out is also included. These notes assume no prior knowledge of Calculus. Solving the Heat Equation — In this section we go through the complete separation of variables process, including solving the two ordinary differential equations the process generates.

Each cheat sheets comes in two versions. I have included a couple of topics that are not that important to a Calculus class, but students do seem to have trouble with on occasion. We will also give brief overview on using Laplace transforms to solve nonconstant coefficient differential equations.

The purpose of this document is go a little beyond what most people see when the first are introduced to complex numbers in say a College Algebra class. However, anyone needing a review of some of the basic algebra, trig, exponential functions and logarithms should find the information of use.

It is also assumed that you have a fairly good knowledge of Trig. The Definition — In this section we give the definition of the Laplace transform. How To Study Math - This is a short section with some advice on how to best study mathematics.

We define fundamental sets of solutions and discuss how they can be used to get a general solution to a homogeneous second order differential equation.

We give a detailed examination of the method as well as derive a formula that can be used to find particular solutions. Nonhomogeneous Differential Equations — In this section we will discuss the basics of solving nonhomogeneous differential equations.

Because I wanted to make this a fairly complete set of notes for anyone wanting to learn differential equations have included some material that I do not usually have time to cover in class and because this changes from semester to semester it is not noted here.

Real Eigenvalues — In this section we will solve systems of two linear differential equations in which the eigenvalues are distinct real numbers. Heat Equation with Non-Zero Temperature Boundaries — In this section we take a quick look at solving the heat equation in which the boundary conditions are fixed, non-zero temperature.

We will also do a few more interval of validity problems here as well. If your device is not in landscape mode many of the equations will run off the side of your device should be able to scroll to see them and some of the menu items will be cut off due to the narrow screen width.

Higher Order Differential Equations - In this chapter we will look at extending many of the ideas of the previous chapters to differential equations with order higher that 2nd order.

Power Series — In this section we give a brief review of some of the basics of power series. This will include illustrating how to get a solution that does not involve complex numbers that we usually are after in these cases.

As we will see they are mostly just natural extensions of what we already know who to do. Definitions — In this section some of the common definitions and concepts in a differential equations course are introduced including order, linear vs. Graphing particular types of equations is covered extensively in the notes, however, it is assumed that you understand the basic coordinate system and how to plot points.

Differential Equations Here are my notes for my differential equations course that I teach here at Lamar University. This will include deriving a second linearly independent solution that we will need to form the general solution to the system.

Solutions to Systems — In this section we will a quick overview on how we solve systems of differential equations that are in matrix form.

Partial Differential Equations - In this chapter we introduce Separation of Variables one of the basic solution techniques for solving partial differential equations. We define the characteristic polynomial and show how it can be used to find the eigenvalues for a matrix.

Eigenvalues and Eigenfunctions — In this section we will define eigenvalues and eigenfunctions for boundary value problems. These downloadable versions are in pdf format.

We will also work a few examples illustrating some of the interesting differences in using boundary values instead of initial conditions in solving differential equations.Welcome to my online math tutorials and notes. The intent of this site is to provide a complete set of free online (and downloadable) notes and/or tutorials for classes that I teach at Lamar University.I've tried to write the notes/tutorials in such a way that they should be accessible to anyone wanting to learn the subject regardless of whether you are in my classes or not.

Linear Equations – In this section we solve linear first order differential equations, i.e. differential equations in the form \(y' + p(t) y = g(t)\). We give an in depth overview of the process used to solve this type of differential equation as well as a derivation of the formula needed for the integrating factor used in the solution process.

DownloadWrite a logarithmic equation in exponential form

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