factorial, writtenis– we multiply together all the whole numbers between 1 andinclusive. Sometimes we have to multiply or divide factorial expressions:. Ifthen A common expression isWe often need to divide expressions of the form The above expression may be all you need, but more terms may be cancelled, though you may not think the resulting […]

## Evaluating Expressions Given in Terms of Roots of Quadratic Equations

If a quadratic equationhas rootsandthen we can write the equation asExpanding the brackets gives (1) Suppose that we have the equationwith rootsandthen comparing with (1) givesand We can use simple algebra to evaluate many expressions given in terms ofandwithout ever knowing what the values ofand Example: Example: Example: Example: Example: Any function symmetric in terms […]

## Decomposing Transformations I

Matrices representing transformations may be multiplied by each other, representing a composition of transformations. We also need to be able to express a matrix as a product of matrices representing transformations. The transformations are are particularly interested in are enlargements or scalings, rotations and reflections. Suppose then that we have the matrix This matrix has […]

## Recurrence Relations and Closed Forms

A recurrence relation uses each term or maybe several terms in a sequence to calculate succeeding terms. If the nth term is denoted by u-n then theterm is denoted byUsing this notation, an example of a recurrence relation is given by If the first term of this sequence is 6 The second term The third […]

## Transforming a Non – Separable Differential Equation into a Separable Differential Equation

Separating the variables to solve differential equations is a familiar and simple method, but limited in it’s usefulness because most equations are not separable. If however the equation is of the form whereandare both of the formthen separability can be achieved with the substitution Proof: ifthen Simplification of the right hand side returns Now x […]

## Roots of Polynomial Equations With Real Coefficients

If a polynomial equationhas real coefficients, and ifis a root, so thatthen the complex conjugate ofis also a root so that Proof: Ifis a root so thatthen taking the complex conjugate of both sides gives sincefor henceis also a root. This means that given a complex rootwe can write down two factorsandmultiply them together to […]

## The Definition of Constant Coefficient Linear Homogeneous, Linear Non Homogeneous, Non Linear Homogeneous and Non Linear Non Homogeneous Second Order Differential Equations

Constant Coefficient Linear Homogeneous Differential Equations Constant Coefficient Linear Non Homogeneous Differential Equations Constant Coefficient Non Linear Homogeneous Differential Equations In the above equation there is at least one product or power of or Constant Coefficient Non Homogeneous Second Order Differential Equations In the above equation there is at least one product or power of […]

## Integration in Polar Coordinates

In rectangular coordinates we find the area bounded by the curvethe x-axis, and the ordinates atandusing The corresponding problem in polar coordinates is that of determining the area bounded by the curveand the two radius vectorsandIn Fig. 4 this is the area bounded by the curve and the lines OA and OB. We divide the-interval […]

## Finding the Area Between Two Curves Written in Polar Coordinates

Problems of this kind are variable and each one may demand a different approach. In the diagram belowis the curveandis the curveW e have to find the shaded are and this must be done in several stages. We find the points A and B of intersection of the two curves, specifically, we find thevalues andHaving […]

## Solving Second Order Linear Non – Homogeneous Differential Equations

Any differential equation of the form(1) is a second order differential equations and there is a standard technique for solving any equation of this sort. We solve the homogeneous equation(2) first for the ‘complementary’ solutionWe assume a solution of the formand substitute this into (2). We extract the non zero factor – since no exponential […]