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. Basic Concepts In this chapter we will be looking exclusively at linear second order differential equations. The most general linear second order differential equation is in the form.

Substitute this value of y in equation 4. This will give you an equation in z. Substitute this value of z in equation 6 and solve for y.

Substitute 3 for y and 2 for z in equation 1 and solve for x. The process of elimination involves several steps: First you reduce three equations to two equations with two variables, and then to one equation with one variable. Decide which variable you will eliminate.

It makes no difference which one you choose. Let us eliminate y first. Add equation 1 to equation 2 to form Equation 4.

Add 3 times equation 2 to 5 times equation 3 to form equation 5. We now have two equations with two variables. Multiply both sides of equation 4 by and add the transformed equation 4 to equation 5 to create equation 6 with just one variable. Solve for x in equation 5.

Substitute 1 for x in equation 5 and solve for z. Substitute 1 for x and 2 for z in equation 1 and solve for y. The process of using matrices is essentially a shortcut of the process of elimination. Each row of the matrix represents an equation and each column represents coefficients of one of the variables.

Create a three-row by four-column matrix using coefficients and the constant of each equation. The vertical lines in the matrix stands for the equal signs between both sides of each equation.

The first column contains the coefficients of x, the second column contains the coefficients of y, the third column contains the coefficients of z, and the last column contains the constants.

We want to convert the original matrix to the following matrix. We work with column 1 first. Multiply Row 1 by to form a new Row 1. We will now complete with column 1.

We want zeros in Cell 21 and Cell We achieve this by adding -3 times Row 1 to Row 2 to form a new Row 2 and, by adding -5 times Row 1 to Row 3 to form a new Row 3. Let's now work with column 2. We want the number 1 in Cell We do this by multiplying row 2 by to form a new row 2.

To complete our work with Row 2 we want zeros in cell 12 and cell Let's now work the Column 3. We first want the number 1 in Cell We can do this by multiplying Row 3 by to form a new Row 3. Let's now put zeros in Cell 13 and Cell We do this by adding times Row 3 to Row 1 to form a new Row 1, and by adding times Row 3 to Row 2 to form a new Row 2.

The solution can now be ready off the last matrix: If you would like to go back to the problem page, click on Problem. If you would like to review the solution to the next problem, click on Problem If you would like to return to the beginning of the three by three system of equations, click on Example.

This site was built to accommodate the needs of students. The topics and problems are what students ask for. We ask students to help in the editing so that future viewers will access a cleaner site.Find a homogeneous linear system of two equations in three unknowns whose solution space consists of those vectors in R3 that are orthogonal to a=(1,1,1) and b=(-2,3,0) Ask for details Follow.

SYSTEMS OF EQUATIONS in THREE VARIABLES. It is often desirable or even necessary to use more than one variable to model a situation in many fields.

When this is the case, we write and solve a system of equations in order to answer questions about the situation. The solution is: and z=2. CBE, Levicky 2 iv). In example (iii) above, how would you classify the changes in the gas tank?

Are these changes indicative of a batch, . 2> Let a and b represent real numbers. Describe the possible solution sets of the (linear) equation ax = b (hint: the number of solutions depends upon a and b).

3> Construct threee different augmented matrices for linear systems whose solution set is x1 = 3, x2= -2, x3= The origins of algebra can be traced to the ancient Babylonians, [page needed] who developed a positional number system that greatly aided them in solving their rhetorical algebraic equations.

The Babylonians were not interested in exact solutions but approximations, and so they would commonly use linear interpolation to approximate intermediate values.

Thus, the solution is x = 3, y = 2; or (3, 2). Notice that we are simply applying the addition property of equality so we can obtain an equation containing a single variable. The equation in one variable, together with either of the original equations, then forms an equivalent system whose solution is easily obtained.

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Differential Equations - Basic Concepts