Solving Second Order Linear Homogeneous Differential Equations

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 […]

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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 […]

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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 […]

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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 […]

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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 […]

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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 […]

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