We will also derive from the complex roots the standard solution that is typically used in this case that will not involve complex numbers.

The Wave Equation — In this section we do a partial derivation of the wave equation which can be used to find the one dimensional displacement of a vibrating string. We also give a nice relationship between Heaviside and Dirac Delta functions. How To Study Math - This is a short section with some advice on how to best study mathematics.

Sometimes questions in class will lead down paths that are not covered here. There is some review of a couple of Algebra and Trig topics, but for the most part it is assumed that you do have a decent background in Algebra and Trig.

Series Solutions — In this section we are going to work a quick example illustrating that the process of finding series solutions for higher order differential equations is pretty much the same as that used on 2nd order differential equations.

Mechanical Vibrations — In this section we will examine mechanical vibrations. The point of this section is only to illustrate how the method works. Note that this is in contrast to the previous section when we generally required the boundary conditions to be both fixed and zero.

We also show the formal method of how phase portraits are constructed. Vibrating String — In this section we solve the one dimensional wave equation to get the displacement of a vibrating string.

Fundamental Sets of Solutions — In this section we will a look at some of the theory behind the solution to second order differential equations. 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 define the characteristic polynomial and show how it can be used to find the eigenvalues for a matrix.

The Definition — In this section we give the definition of the Laplace transform. Undetermined Coefficients — In this section we work a quick example to illustrate that using undetermined coefficients on higher order differential equations is no different that when we used it on 2nd order differential equations with only one small natural extension.

In addition, we also give the two and three dimensional version of the wave equation. We will also give and an alternate method for finding the Wronskian. Partial Differential Equations - In this chapter we introduce Separation of Variables one of the basic solution techniques for solving partial differential equations.

Intervals of Validity — In this section we will give an in depth look at intervals of validity as well as an answer to the existence and uniqueness question for first order differential equations.

Nonhomogeneous Systems — In this section we will work quick examples illustrating the use of undetermined coefficients and variation of parameters to solve nonhomogeneous systems of differential equations.

Undetermined Coefficients — In this section we introduce the method of undetermined coefficients to find particular solutions to nonhomogeneous differential equation. We also derive the formulas for taking the Laplace transform of functions which involve Heaviside functions.

Here is a complete listing of all the subjects that are currently available on this site as well as brief descriptions of each. Modeling with First Order Differential Equations — In this section we will use first order differential equations to model physical situations.

In particular we will look at mixing problems modeling the amount of a substance dissolved in a liquid and liquid both enters and exitspopulation problems modeling a population under a variety of situations in which the population can enter or exit and falling objects modeling the velocity of a falling object under the influence of both gravity and air resistance.

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.

Systems of Equations — In this section we will give a review of the traditional starting point for a linear algebra class.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.

DownloadHow to write an exponential equation into logarithmic form

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