Due to the nature of the mathematics on this site it is best views in landscape mode. 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. Differential Equations Here are my notes for my differential equations course that I teach here at Lamar University. 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.
Due to the nature of the mathematics on this site it is best views in landscape mode. 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.
There will not be a lot of details in this section, nor will we be working large numbers of examples.
The first special matrix is the square matrix. In other words, it has the same number of rows as columns. In a square matrix the diagonal that starts in the upper left and ends in the lower right is often called the main diagonal.
The next two special matrices that we want to look at are the zero matrix and the identity matrix.
Here are the general zero and identity matrices. These are matrices that consist of a single column or a single row. Arithmetic We next need to take a look at arithmetic involving matrices. If it is true, then we can perform the following multiplication.
Here are a couple of the entries computed all the way out. Determinant The next topic that we need to take a look at is the determinant of a matrix. The determinant is actually a function that takes a square matrix and converts it into a number.
The actual formula for the function is somewhat complex and definitely beyond the scope of this review. The main method for computing determinants of any square matrix is called the method of cofactors.
We can give simple formulas for each of these cases. There is an easier way to get the same result. A quicker way of getting the same result is to do the following.
First write down the matrix and tack a copy of the first two columns onto the end as follows. What we do is multiply the entries on each diagonal up and the if the diagonal runs from left to right we add them up and if the diagonal runs from right to left we subtract them.
Here is the work for this matrix. Matrix Inverse Next, we need to take a look at the inverse of a matrix. Example 4 Find the inverse of the following matrix, if it exists.
In other words, we want a 1 on the diagonal that starts at the upper left corner and zeroes in all the other entries in the first three columns. If you think about it, this process is very similar to the process we used in the last section to solve systems, it just goes a little farther.
Here is the work for this problem. Example 5 Find the inverse of the following matrix, provided it exists. However, there is no way to get a 1 in the second entry of the second column that will keep a 0 in the second entry in the first column.
We will leave off this discussion of inverses with the following fact. Systems of Equations Revisited We need to do a quick revisit of systems of equations. The solving process is identical.
There will be no solutions. There will be exactly one solution. There will be infinitely many solutions. In fact, we can go a little farther now. This gives the following fact. We also saw linear independence and linear dependence back when we were looking at second order differential equations.
In that section we were dealing with functions, but the concept is essentially the same here.
Example 6 Determine if the following set of vectors are linearly independent or linearly dependent.Now we have the 2 equations as shown below. Notice that the \(j\) variable is just like the \(x\) variable and the \(d\) variable is just like the \(y\).
It’s easier to put in \(j\) and \(d\) so we can remember what they stand for when we get the answers. This is what we call a system, since we have to solve for more than one variable – we have to solve for 2 here. In this section we will give a brief review of matrices and vectors.
We will look at arithmetic involving matrices and vectors, finding the inverse of a matrix, computing the determinant of a matrix, linearly dependent/independent vectors and converting systems of equations into matrix form.
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. We can graph the set of parametric equations above by using a graphing calculator.
First change the MODE from FUNCTION to PARAMETRIC, and enter the equations . lausannecongress2018.com Solve word problems leading to inequalities of the form px + q > r or px + q. You could start solving this system by adding down the columns to get 4y = 4.
You can do something similar with lausannecongress2018.com instance, given the following matrix.