Any differential equation of the formis a second order differential equations and there is a standard technique for solving any equation of this sort. We assume a solution of the formand substitute this into the equation. We extract the non zero factor – since no exponential is zero for any finite x –to obtain a […]

## More Advanced Inequalities

Any linear inequality of the formcan be solved just like a linear equation, by moving terms and changing signs for those terms that move to the other side of an equals sign: It must be clear that when we multiple or divide by a negative number the inequality changes direction. Inequalities can be much more […]

## Transforming Differential Equations

Not all differential equations take a standard form which can be solved used a standard result. Most differential equations can only be solved numerically on a computer, to a certain precision, which is always a compromise with the time and computing power available and the required precision. Finding the transformation to use is a matter […]

## The Fundamental Theorem of Algebra

The Fundamental Theorem of Algebra is one of the most important theorems in algebra, maybe the most important. It states that the number of roots of a polynomial of degree is exactlyThis means also that the number of roots of the polynomial equationis alsoThis is because ifis a root thenis a factor, so that the […]

## Solving Differential Equations – The Integrating Factor Method

Any equation of the form(1) might be solved using the integrating factor method. This method finds a function ofthat the left hand side can be multiplied by so that the left hand side can be writtenThe integral of this is justso if we can find a function h(x) we can write the solution down as […]

## Properties of Summation

Summation is Linear: These properties may be used to expand terms and separate into different summations: Summation is sequential – you can combine the indices in a natural way: Reversing the indices has no effect – unlike as when the limits of integration are reversed: and These reflect that the order in which numbers are […]

## Parametric Coordinates – Converting Between Rectangular or Cartesian and Parametric Form

Parametric equations define a curve in terms of some third quantity. Theandcoordinates are expressed in terms of this quantity, called a parameter. For example the linewhich is written in cartesian coordinates may be written in parametric form aswhereis the parameter.. Notice that thecoordinate here is always one more than thecoordinate, reflecting that for the linewe […]

## Finding the Equations of Tangents and Normals to Parametric Curves

Parametric curves take the formwhereis a parameter. Each value of gives (not necessarily unique) values ofandIf we need to find the equation of a tangent or normal to the curve at any pointthen we need the gradientat that point. We findusing the identitywill be a function ofso if we have the vale of we can […]

## Series – Standard Expressions for Sums of Powers of Integers

There are several very useful expressions that expression the sum of powers of integers in terms of the range of summation. Notice that the degree of the expression on the right hand side is one more than the terms of the left hand side, so that summing a polynomial sequence increases the degree by one. […]

## Series – Standard Expressions for Sums of Powers of Integers

There are several very useful expressions that expression the sum of powers of integers in terms of the range of summation. Notice that the degree of the expression on the right hand side is one more than the terms of the left hand side, so that summing a polynomial sequence increases the degree by one. […]