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Integration of Ordinary Differential Equations part 1

Problems involving ordinary differential equations (ODEs) can always be reduced to the study of sets of first-order differential equations. For example the second-order equation dy d2 y = r(x) + q(x) 2 dx dx can be rewritten as two first-order equations dy = z(x) dx dz = r(x) − q(x)z(x) dx | Chapter 16. Integration of Ordinary Differential Equations 16.0 Introduction Problems involving ordinary differential equations ODEs can always be reduced to the study of sets of first-order differential equations. For example the second-order equation 2 q x r x 16.0.1 dx2 dx can be rewritten as two first-order equations dy dx z x dz dx r x q x z x 16.0.2 where z is a new variable. This exemplifies the procedure for an arbitrary ODE. The usual choice for the new variables is to let them be just derivatives of each other and of the original variable . Occasionally it is useful to incorporate into their definition some other factors in the equation or some powers of the independent variable for the purpose of mitigating singular behavior that could result in overflows or increased roundoff error. Let common sense be your guide If you find that the original variables are smooth in a solution while your auxiliary variables are doing crazy things then figure out why and choose different auxiliary variables. The generic problem in ordinary differential equations is thus reduced to the study of a set of N coupled first-order differential equations for the functions yi i 1 2 . . N having the general form dy x fi x yi . . yN i 1 . N 16.0.3 dx Sample page from NUMERICAL RECIPES IN C THE ART OF SCIENTIFIC COMPUTING ISBN 0-521-43108-5 where the functions fi on the right-hand side are known. A problem involving ODEs is not completely specified by its equations. Even more crucial in determining how to attack the problem numerically is the nature of the problem s boundary conditions. Boundary conditions are algebraic conditions 707 708 Chapter 16. Integration of Ordinary Differential Equations on the values of the functions y in 16.0.3 . In general they can be satisfied at discrete specified points but do not hold between those points i.e. are not preserved automatically by the differential equations. Boundary conditions can be as simple as requiring that certain variables have .