• 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 of that the left hand side can be multiplied by so that the left hand side can be written The integral of this is just so if we can find a...

  • The Integrating Factor Method of Solving First Order Differential Equations

    gives where Some first order differential equations are not separable. Often the most suitable way to solve it is the integrating factor method, which can be used to solve equations of the form If we multiply both sides by the integrating factor, the...

  • Improper Integrals

    An improper integral is one where either of the following holds one of the integrals is or or the limits are and and are all improper integrals. the integrand (the function being integrated) includes a term evaluated at one or both limits which takes...

  • Integral of Reciprocals of Linear Expressions by Rearrangement

    Integrating any quotient of the form can be done by making the substitution For example to find sub so the integral becomes (1) Complete the integral by substituting to obtain There is a method that is often quicker. The method is to take out factors...

  • Solving Differential Equations Using Perturbation Expansions

    the solution to the unperturbed equation, Substitute this expression for into (2) to obtain This can be solved by the integrating factor method to obtain Substitution of the expressions for and into the third of these gives This can again be solved by...

  • Differential Equations - Separating Variables

    It is a very unusual thing to be given a differential equation that will just, well, integrate. Usually some manipulations must be performed, whether it is simplifying, grouping like terms, simplifying, making substitutions or separating variables –...

  • Differential Equatioins - Separating Variables

    It is a very unusual thing to be given a differential equation that will just, well, integrate. Usually some manipulations must be performed, whether it is simplifying, grouping like terms, simplifying, making substitutions or separating variables –...

  • Transforming and Solving Non - Linear, Non - Homogeneous Differential Equations

    be transformed into a simpler form and solved using the one of standard techniques of separation of variables, the integrating factor method or the procedure for solving linear equations. Example: (1). Use the transformation and find the general...

  • Derivation of the Poisson Distribution

    gives so that In general So As we obtain Now prove by induction. Assume and use from above as Now multiply by the integrating factor to give Integration now gives

  • Modelling Raindrops - Mass Increasing at Constant Rate

    respect to x then and so Write The equation becomes Now divide by and write : We can solve this equation using the integrating factor method: The equation now becomes Hence If when then

  • Partial Fractions

    An algebraic fraction is any expression of the form where and are sums or products of polynomials or both. An expression of this sort typically needs to be written in terms of it's partial fractions – where is written as a sum of algebraic fractions -...

  • The Laplace Transform

    The Laplace transform presents an alternative method of solving differential equations. It enables us to transform a differential equation into an algebraic equation. We can solve the algebraic equation and apply an inverse transformation to obtain the...

  • Normalizing the Wavefunction

    The Born interpretation gives the probability of finding a particle with wavefunction - I have shown the wavefunction here to be a function of here, though I need not have done and do not use this below - in the volume of space between and is The...

  • Natural Logairthms

    One number raised to the power of another is called a base. In the expression 3^4 the base is 3. The most common base is 10 – we count and measure things in multiples of 10 because we have 10 fingers on which to count. There is however, one base which...

  • The Operational Amplifier

    Operational amplifiers are linear devices that have all the properties required for nearly ideal DC Voltage amplification, used extensively in signaling circuits, filtering or to perform mathematical operations such as addition and subtraction.. An...

  • Electric Displacement

    Dielectric may not be neutral even when unpolarised. If the dielectric carries a charge density of free charges representing a net surplus or deficit of electrons in the atoms of the dielectric and is the charge density due to the polarizing effect of...

  • Properties of the Eigenfunctions

    The eigenfunctions are the solutions of the eigenfunction equation the solutions for the one dimensional simple harmonic oscillator case, are polynomials in multiplied by a gaussian If the are normalized to unity they have the following properties: The...


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