We define the characteristic polynomial and show how it can be used to find the eigenvalues for a matrix. This includes a working knowledge of differentiation and integration.

We will also do a few more interval of validity problems here as well. Periodic Functions and Orthogonal Functions — In this section we will define periodic functions, orthogonal functions and mutually orthogonal functions.

As we will see this is exactly the equation we would need to solve if we were looking to find the equilibrium solution i. Eigenvalues and Eigenfunctions — In this section we will define eigenvalues and eigenfunctions for boundary value problems.

This portion of the site should be of interest to anyone looking for common math errors. That will be done in later sections. We will develop of a test that can be used to identify exact differential equations and give a detailed explanation of the solution process.

Note that this is in contrast to the previous section when we generally required the boundary conditions to be both fixed and zero. In other words, given a Laplace transform, what function did we originally have?

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 will use reduction of order to derive the second solution needed to get a general solution in this case.

Direction Fields — In this section we discuss direction fields and how to sketch them. We will also give brief overview on using Laplace transforms to solve nonconstant coefficient differential equations. Trig Cheat Sheets - Here is a set of common trig facts, properties and formulas.

Welcome to my online math tutorials and notes. There are four different cheat sheets here. As we will see they are mostly just natural extensions of what we already know who to do.

Bernoulli Differential Equations — In this section we solve linear first order differential equations, i. First Order Differential Equations - In this chapter we will look at several of the standard solution methods for first order differential equations including linear, separable, exact and Bernoulli differential equations.

Modeling with First Order Differential Equations — In this section we will use first order differential equations to model physical situations. However, with Differential Equation many of the problems are difficult to make up on the spur of the moment and so in this class my class work will follow these notes fairly close as far as worked problems go.

We work a couple of examples of solving differential equations involving Dirac Delta functions and unlike problems with Heaviside functions our only real option for this kind of differential equation is to use Laplace transforms. Phase Plane — In this section we will give a brief introduction to the phase plane and phase portraits.

Here are a couple of warnings to my students who may be here to get a copy of what happened on a day that you missed. We will also define the odd extension for a function and work several examples finding the Fourier Sine Series for a function.

The advantage of starting out with this type of differential equation is that the work tends to be not as involved and we can always check our answers if we wish to. We derive the characteristic polynomial and discuss how the Principle of Superposition is used to get the general solution.

This table gives many of the commonly used Laplace transforms and formulas. Exact Equations — In this section we will discuss identifying and solving exact differential equations. We give a detailed examination of the method as well as derive a formula that can be used to find particular solutions.

We do not, however, go any farther in the solution process for the partial differential equations. Several topics rely heavily on trig and knowledge of trig functions. Eigenvalues and Eigenvectors — In this section we will introduce the concept of eigenvalues and eigenvectors of a matrix.

Nonhomogeneous Differential Equations — In this section we will discuss the basics of solving nonhomogeneous differential equations. We will give a derivation of the solution process to this type of differential equation. Here is a listing and brief description of the material that is in this set of notes.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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