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Second Order Differential Equations

Second OrderDifferential Equations this Section we start to learn how to solve Second Order Differential Equations of a particular type:those that are linear and have constant coefficients. Such Equations are used widely in the modellingof physical phenomena, for example, in the analysis of vibrating systems and the analysis of solution of these Equations is achieved in stages. The first stage is to find what is called a com-plementary function . The Second stage is to find a particular integral . Finally, the complementaryfunction and the particular integral are combined to form the general solution. PrerequisitesBefore starting this Section you should.. understand what is meant by a differentialequation understand complex numbers (10)'&$%Learning OutcomesOn completion you should be able to.. recognise a linear, constant coefficientequation understand what is meant by the terms auxiliary equation and complementaryfunction find the complementary function when theauxiliary equation has real, equal or complexroots30 HELM (2008):Workbook 19: Differential Equations 1.

1. Constant coefficient second order linear ODEs We now proceed to study those second order linear equations which have constant coefficients. The general form of such an equation is: a d2y dx2 +b dy dx +cy = f(x) (3) where a,b,c are constants. The homogeneous form of (3) is the case when f(x) ≡ 0: a d2y dx2 +b dy dx +cy = 0 (4)

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